[{"data":1,"prerenderedAt":1038},["ShallowReactive",2],{"layer:hcf-and-lcm:understand":3},{"layer":4,"contentHash":1018,"dependencyHashes":1019,"approval":1032,"releaseId":1037},{"schemaVersion":5,"conceptId":6,"locale":7,"depth":8,"revision":5,"title":9,"subtitle":10,"summary":11,"objectives":12,"estimatedMinutes":18,"plate":19,"blocks":40,"sourceIds":1013,"reviewStatus":1014,"authoring":1015},1,"hcf-and-lcm","en","understand","Four ways to find the HCF and LCM","Listing, prime factors, long division and the ladder, and why they work","Precise definitions, then four methods: listing, prime factorisation with a Venn picture, long (continued) division for the HCF and common division for the LCM. Three numbers, the rule HCF × LCM = product, co-primes, fractions and the classic mix-ups.",[13,14,15,16,17],"Find the HCF and LCM of two or three numbers by listing, prime factorisation, long division and common division.","Explain the prime-factor rules (smallest powers for HCF, highest powers for LCM) using a Venn diagram.","Use HCF × LCM = product of two numbers to find a missing number, and know why it fails for three.","Use the HCF to simplify fractions and the LCM to add and subtract them.","Spot and correct common mistakes in HCF and LCM working.",45,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Understand",{"label":26,"value":27},"Reading time","≈ 45 minutes",{"label":29,"value":30},"Prior knowledge","Factors, multiples, primes (Discover)",{"label":32,"value":33},"Chapters","13",{"label":35,"value":36},"Methods","Listing, primes, division, ladder",{"label":38,"value":39},"Labs","Factor trees, Venn, division, three numbers, match, sort",[41,45,51,57,62,66,69,122,127,140,145,148,153,156,178,199,204,209,212,216,237,248,253,268,273,276,307,317,329,334,337,349,376,400,412,417,420,431,435,440,443,454,477,488,493,496,518,527,545,550,553,565,579,591,596,600,604,608,613,646,701,710,715,724,734,747,757,761,765,769,778,787,799,810,822,834,839,966,982,986,992,997,1002],{"id":42,"type":43,"markdown":44},"intro-understand","prose","In Discover you found the HCF and LCM by **listing**: write out factors or multiples and look for the ones that match. That works well for small numbers. But try listing the multiples of 84 and 126 until they meet, or all the factors of 1,080, and you will soon want something better.\n\nThis layer gives you **four methods** and the reasons they work:\n\n1. **Listing** factors or multiples: good for small numbers and for seeing the idea.\n2. **Prime factorisation**, with a Venn diagram in your head: the method that shows *why*.\n3. **Long division** (continued division) for the HCF: fast for big numbers.\n4. **Common division** (the ladder) for the LCM: tidy for three or more numbers.\n\nAlong the way you will meet three facts that make HCF and LCM problems much quicker, the classic mix-ups, and the place where all this matters most in school maths: **fractions**.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"how-to-read-u","callout","observation","How to use this layer","Every method is followed by a worked example and a lab. Try each method on paper at least once before you use the lab to check. By the end, pick your favourite method, but make sure you can use them all: exams often ask for a particular one.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch01","chapter","Precise definitions and notation","Chapter 01","1 Definitions",{"id":58,"type":47,"variant":59,"title":60,"markdown":61},"def-hcf-precise","definition","Highest Common Factor","The **Highest Common Factor** of two or more whole numbers is the **largest** whole number that divides **each** of them exactly.\n\nNotation: **HCF(a, b)**. Other names: **GCD** (Greatest Common Divisor), **GCF** (Greatest Common Factor). Many calculators and computer languages call it *gcd*.",{"id":63,"type":47,"variant":59,"title":64,"markdown":65},"def-lcm-precise","Lowest Common Multiple","The **Lowest (Least) Common Multiple** of two or more whole numbers is the **smallest positive** whole number that **each** of them divides exactly.\n\nNotation: **LCM(a, b)**. Computer languages often call it *lcm*.",{"id":67,"type":43,"markdown":68},"def-unpack","Read those definitions slowly, because every word matters.\n\n- **\"Each\"** means all of them. A common factor of 12, 18 and 20 must divide all three. 3 divides 12 and 18 but not 20, so it is not a common factor of the three.\n- **\"Exactly\"** means remainder 0.\n- **\"Positive\"** in the LCM rules out 0, which is a multiple of every number.\n- **\"Divides\"** has a direction. \"4 divides 12\" means 12 ÷ 4 is exact. We sometimes write this as **4 | 12**, read \"4 divides 12\". So HCF(a, b) | a, HCF(a, b) | b, a | LCM(a, b) and b | LCM(a, b).\n\nThe order of the numbers never matters: HCF(12, 18) = HCF(18, 12), and the same for LCM.",{"id":70,"type":71,"title":72,"terms":73},"glossary-understand","glossary","Vocabulary for the methods",[74,78,82,86,90,94,98,102,106,110,114,118],{"term":75,"meaning":76,"example":77},"prime number","A whole number greater than 1 with exactly two factors: 1 and itself.","2, 3, 5, 7, 11, 13, …",{"term":79,"meaning":80,"example":81},"composite number","A whole number greater than 1 with more than two factors.","4, 6, 8, 9, 10, 12, …",{"term":83,"meaning":84,"example":85},"prime factor","A factor of a number that is itself prime.","The prime factors of 12 are 2 and 3.",{"term":87,"meaning":88,"example":89},"prime factorisation","Writing a number as a product of primes only.","60 = 2 × 2 × 3 × 5",{"term":91,"meaning":92,"example":93},"index form","Prime factorisation written with powers (indices) for repeated primes.","60 = 2² × 3 × 5",{"term":95,"meaning":96,"example":97},"power \u002F index \u002F exponent","The small raised number saying how many times a factor is repeated.","In 2³ the index is 3: 2 × 2 × 2.",{"term":99,"meaning":100,"example":101},"factor tree","A branching diagram that splits a number into factor pairs until every branch ends in a prime.","36 → 6 × 6 → 2 × 3 × 2 × 3",{"term":103,"meaning":104,"example":105},"co-prime (relatively prime)","Two numbers whose only common factor is 1, so their HCF is 1.","8 and 15 are co-prime.",{"term":107,"meaning":108,"example":109},"dividend, divisor, quotient, remainder","In 38 ÷ 5 = 7 remainder 3: 38 is the dividend, 5 the divisor, 7 the quotient, 3 the remainder.","38 = 5 × 7 + 3",{"term":111,"meaning":112,"example":113},"continued division","The long division method for HCF: divide, then divide the divisor by the remainder, again and again, until the remainder is 0.","HCF(252, 198) = 18",{"term":115,"meaning":116,"example":117},"common division (ladder)","The LCM method that divides all the numbers together by primes, row by row.","LCM(12, 18, 30) = 180",{"term":119,"meaning":120,"example":121},"Venn diagram","Overlapping circles showing what two groups share; here, shared prime factors go in the overlap.","HCF = product of the overlap",{"id":123,"type":53,"title":124,"eyebrow":125,"navLabel":126},"ch02","Method 1: listing, and where it runs out","Chapter 02","2 Listing",{"id":128,"type":129,"title":130,"problem":131,"steps":132,"help":138},"we-listing","worked_example","HCF and LCM of 20 and 30 by listing","Find HCF(20, 30) and LCM(20, 30) by listing.",[133,134,135,136,137],"Factors of 20: 1, 2, 4, 5, 10, 20.","Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30.","Common factors: 1, 2, 5, 10. **HCF = 10**.","Multiples of 30 (the bigger number): 30, 60, 90, 120, …","Check each against 20: 30 ✗, 60 ✓ (60 = 20 × 3). **LCM = 60**.",{"simplerExplanation":139},"For the LCM, list multiples of the **bigger** number and stop at the first one the smaller number divides. It is much faster than listing both.",{"id":141,"type":47,"variant":142,"title":143,"markdown":144},"aha-bigger-first","aha","Always skip-count the bigger number","To find LCM(9, 12), don't write out both tables. Count in 12s and test each against 9: 12 ✗, 24 ✗, **36** ✓. Three tries.\n\nCounting in 9s would take four tries (9, 18, 27, 36) and you would still have to check each against 12. The bigger the gap between the numbers, the more time this trick saves.",{"id":146,"type":43,"markdown":147},"listing-limits","Listing has two weaknesses.\n\n- **It gets slow.** LCM(84, 126) = 252. Counting in 126s, you need 2 tries, which is fine. But LCM(84, 125) = 10500: you would need 84 multiples of 125 before one works!\n- **It is easy to miss a factor.** 360 has 24 factors. Miss one and you might miss the HCF.\n\nThe next methods build the answer directly, instead of searching for it.",{"id":149,"type":53,"title":150,"eyebrow":151,"navLabel":152},"ch03","Prime factorisation: every number’s recipe","Chapter 03","3 Prime recipes",{"id":154,"type":43,"markdown":155},"pf-prose","Every whole number bigger than 1 is either prime or can be built by multiplying primes. Its **prime factorisation** is its recipe:\n\n- 36 = 2 × 2 × 3 × 3 = **2² × 3²**\n- 48 = 2 × 2 × 2 × 2 × 3 = **2⁴ × 3**\n- 60 = 2 × 2 × 3 × 5 = **2² × 3 × 5**\n- 84 = 2 × 2 × 3 × 7 = **2² × 3 × 7**\n\nHowever you split a number, you always end up with the **same** primes the same number of times. (Split 36 as 4 × 9 or as 6 × 6 or as 2 × 18: you always reach two 2s and two 3s.) This fact, that the recipe is unique, is called the **Fundamental Theorem of Arithmetic**. It is the reason the prime-factor method for HCF and LCM works, and the Deepen layer explains why it is true.",{"id":157,"type":158,"component":159,"componentVersion":5,"config":160,"objective":171,"textAlternative":172,"help":173},"lab-factor-tree","interactive","factor-tree",{"numbers":161,"showIndexForm":170},[162,163,164,165,166,167,168,169],36,48,60,84,72,120,180,252,true,"Build factor trees and write each number’s prime factorisation in index form, ready for HCF and LCM.","In this lab you split a number into two factors, then split each factor again, until every branch ends in a prime. The primes at the ends of the branches, multiplied together, give the prime factorisation.\n\nThe numbers and their factorisations:\n- 36 = 2 × 2 × 3 × 3 = **2² × 3²**\n- 48 = 2 × 2 × 2 × 2 × 3 = **2⁴ × 3**\n- 60 = 2 × 2 × 3 × 5 = **2² × 3 × 5**\n- 84 = 2 × 2 × 3 × 7 = **2² × 3 × 7**\n- 72 = 2 × 2 × 2 × 3 × 3 = **2³ × 3²**\n- 120 = 2 × 2 × 2 × 3 × 5 = **2³ × 3 × 5**\n- 180 = 2 × 2 × 3 × 3 × 5 = **2² × 3² × 5**\n- 252 = 2 × 2 × 3 × 3 × 7 = **2² × 3² × 7**\n\nTry splitting 72 two different ways, say 8 × 9 and 6 × 12. The trees look different, but the ends are always three 2s and two 3s.",{"simplerExplanation":174,"hints":175},"Keep breaking the number into two smaller numbers that multiply to it. Stop at primes. Those primes are the recipe.",[176,177],"Start with an easy split: if the number is even, split off 2.","Circle each prime as soon as you reach it so you don’t split it again.",{"id":179,"type":180,"tone":181,"items":182},"spec-index","spec","neutral",[183,187,191,195],{"label":184,"big":185,"value":186},"Long form","2 × 2 × 2 × 3 × 3","Every prime written out: good for Venn diagrams.",{"label":188,"big":189,"value":190},"Index form","2³ × 3²","Repeated primes as powers: good for comparing powers quickly.",{"label":192,"big":193,"value":194},"Value","72","Both are recipes for the same number: 8 × 9 = 72.",{"label":196,"big":197,"value":198},"Divisibility","d | 72","A number divides 72 only if its recipe uses at most three 2s, at most two 3s, and no other prime.",{"id":200,"type":47,"variant":201,"title":202,"markdown":203},"careful-1-not-prime","careful","Do not put 1 in a prime factorisation","1 is **not** a prime number, so it never appears in a prime factorisation. Writing 12 = 1 × 2 × 2 × 3 is not wrong arithmetic, but it is not a prime factorisation, and it can confuse the Venn diagram. If two numbers share no prime at all, the overlap is simply empty, and we say the HCF is 1.",{"id":205,"type":53,"title":206,"eyebrow":207,"navLabel":208},"ch04","HCF by prime factorisation: the shared primes","Chapter 04","4 HCF by primes",{"id":210,"type":43,"markdown":211},"hcf-pf-prose","A common factor of 36 and 48 must be made **only from primes that both recipes contain**, and it cannot use any prime more times than **either** recipe has it.\n\n- 36 = 2 × 2 × 3 × 3\n- 48 = 2 × 2 × 2 × 2 × 3\n\nPair up the primes they share: two 2s are in both (36 has only two), and one 3 is in both (48 has only one). The shared primes are **2 × 2 × 3**, and their product is **12**. That is the HCF.\n\nIn index form: 36 = 2² × 3² and 48 = 2⁴ × 3. Take each **common** prime with its **smaller** power: 2² × 3¹ = 12.",{"id":213,"type":47,"variant":48,"title":214,"markdown":215},"venn-picture","The Venn diagram in your head","Draw two overlapping circles, one for 36 and one for 48. Put the primes in:\n\n- **Overlap (shared):** 2 × 2 × 3\n- **Only 36:** 3\n- **Only 48:** 2 × 2\n\nCheck each circle: left circle = 3 × 2 × 2 × 3 = 36 ✓. Right circle = 2 × 2 × 3 × 2 × 2 = 48 ✓.\n\n**HCF = product of the overlap** = 2 × 2 × 3 = **12**.\n**LCM = product of everything in the picture** = 3 × 2 × 2 × 3 × 2 × 2 = **144**.",{"id":217,"type":158,"component":218,"componentVersion":5,"config":219,"objective":230,"textAlternative":231,"help":232},"lab-venn","hcf-lcm",{"sets":220,"views":228,"challenge":170},[221,222,223,224,226],[162,163],[164,165],[166,167],[18,225],75,[227,18],28,[229],"venn","Drop each prime factor into the right part of a Venn diagram, then read the HCF from the overlap and the LCM from the whole picture.","This lab draws two overlapping circles and places each number’s prime factors in them. Shared primes go in the overlap. The HCF is the product of the overlap; the LCM is the product of every prime in the picture.\n\n- **36 and 48:** 36 = 2² × 3², 48 = 2⁴ × 3. Overlap: 2 × 2 × 3; only 36: 3; only 48: 2 × 2. HCF = **12**, LCM = **144**.\n- **60 and 84:** 60 = 2² × 3 × 5, 84 = 2² × 3 × 7. Overlap: 2 × 2 × 3; only 60: 5; only 84: 7. HCF = **12**, LCM = **420**.\n- **72 and 120:** 72 = 2³ × 3², 120 = 2³ × 3 × 5. Overlap: 2 × 2 × 2 × 3; only 72: 3; only 120: 5. HCF = **24**, LCM = **360**.\n- **45 and 75:** 45 = 3² × 5, 75 = 3 × 5². Overlap: 3 × 5; only 45: 3; only 75: 5. HCF = **15**, LCM = **225**.\n- **28 and 45:** 28 = 2² × 7, 45 = 3² × 5. Overlap: (nothing); only 28: 2 × 2 × 7; only 45: 3 × 3 × 5. HCF = **1**, LCM = **1260**.\n\nFor 28 and 45 the overlap is empty: they are co-prime, the HCF is 1 and the LCM is 28 × 45 = 1,260. In challenge mode you predict the HCF and LCM before the primes are placed.",{"simplerExplanation":233,"hints":234},"Primes both numbers have go in the middle. The middle multiplied together is the HCF. Everything multiplied together is the LCM.",[235,236],"Pair off primes one at a time: a 2 from each number makes one shared 2.","If a prime appears twice in one number and once in the other, only one copy is shared.",{"id":238,"type":129,"title":239,"problem":240,"steps":241},"we-hcf-pf","HCF of 60 and 84 by prime factorisation","Find HCF(60, 84) using prime factorisation.",[242,243,244,245,246,247],"60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.","84 = 2 × 2 × 3 × 7 = 2² × 3 × 7.","Common primes: 2 and 3. (5 is only in 60; 7 is only in 84.)","Smaller power of 2: both have 2², so take 2². Smaller power of 3: both have 3¹, so take 3.","HCF = 2² × 3 = **12**.","Check: 60 ÷ 12 = 5 and 84 ÷ 12 = 7. And 5 and 7 share no factor, so no bigger number divides both. ✓",{"id":249,"type":47,"variant":250,"title":251,"markdown":252},"misconception-highest-power","misconception","\"For the HCF, take the highest powers\"","A very common slip is to take the **highest** power of each prime for the HCF. For 36 = 2² × 3² and 48 = 2⁴ × 3, that would give 2⁴ × 3² = 144, which is bigger than both numbers and cannot divide either. That is the **LCM**.\n\nMemory hook: the HCF has to **fit inside** both numbers, so it can only use what **both** can afford: the **smaller** power.",{"id":254,"type":255,"itemId":256,"prompt":257,"check":258,"hints":262,"feedback":265},"practice-hcf-pf","practice","hcf-and-lcm.understand-hcf-72-120","Given 72 = 2³ × 3² and 120 = 2³ × 3 × 5, what is HCF(72, 120)?",{"kind":259,"answer":260,"tolerance":261},"number",24,0,[263,264],"Which primes are common to both?","Take the smaller power of each common prime.",{"correct":266,"incorrect":267},"Yes: 2³ × 3 = 24. The 5 is not shared, and 3 is shared only once.","Common primes 2 and 3. Smaller powers: 2³ and 3¹. HCF = 8 × 3 = 24.",{"id":269,"type":53,"title":270,"eyebrow":271,"navLabel":272},"ch05","LCM by prime factorisation: every prime, enough times","Chapter 05","5 LCM by primes",{"id":274,"type":43,"markdown":275},"lcm-pf-prose","A common multiple of 36 and 48 must **contain the recipe of 36** and **contain the recipe of 48**. So it needs every prime that appears in either number, each as many times as the **greedier** number needs it.\n\n- 36 needs two 2s and two 3s.\n- 48 needs four 2s and one 3.\n\nThe smallest recipe that satisfies both needs **four 2s** and **two 3s**: 2⁴ × 3² = **144**. That is the LCM.\n\nRule: **LCM = every prime that appears, each with its highest power.**",{"id":277,"type":278,"caption":279,"columns":280,"rows":284},"table-pf-rule","table","The two prime-factor rules side by side",[281,282,283],"Numbers (index form)","HCF: common primes, smallest powers","LCM: all primes, highest powers",[285,289,292,296,300,304],[286,287,288],"36 = 2² × 3²; 48 = 2⁴ × 3","2² × 3 = **12**","2⁴ × 3² = **144**",[290,287,291],"60 = 2² × 3 × 5; 84 = 2² × 3 × 7","2² × 3 × 5 × 7 = **420**",[293,294,295],"72 = 2³ × 3²; 120 = 2³ × 3 × 5","2³ × 3 = **24**","2³ × 3² × 5 = **360**",[297,298,299],"45 = 3² × 5; 75 = 3 × 5²","3 × 5 = **15**","3² × 5² = **225**",[301,302,303],"28 = 2² × 7; 45 = 3² × 5","1 = **1**","2² × 3² × 5 × 7 = **1,260**",[305,287,306],"96 = 2⁵ × 3; 180 = 2² × 3² × 5","2⁵ × 3² × 5 = **1,440**",{"id":308,"type":129,"title":309,"problem":310,"steps":311},"we-lcm-pf","LCM of 72 and 120 by prime factorisation","Find LCM(72, 120).",[312,313,314,315,316],"72 = 2³ × 3²; 120 = 2³ × 3 × 5.","Primes that appear anywhere: 2, 3 and 5.","Highest powers: 2³ (both have 2³), 3² (from 72), 5¹ (from 120).","LCM = 2³ × 3² × 5 = 8 × 9 × 5 = **360**.","Check: 360 ÷ 72 = 5 and 360 ÷ 120 = 3, both exact. ✓",{"id":318,"type":255,"itemId":319,"prompt":320,"check":321,"hints":323,"feedback":326},"practice-lcm-pf","hcf-and-lcm.understand-lcm-60-84","Given 60 = 2² × 3 × 5 and 84 = 2² × 3 × 7, what is LCM(60, 84)?",{"kind":259,"answer":322,"tolerance":261},420,[324,325],"List every prime that appears in either number.","Take the highest power of each.",{"correct":327,"incorrect":328},"Yes: 2² × 3 × 5 × 7 = 420.","Primes 2, 3, 5 and 7. Highest powers 2², 3, 5, 7. LCM = 4 × 3 × 5 × 7 = 420.",{"id":330,"type":53,"title":331,"eyebrow":332,"navLabel":333},"ch06","Method 3: long division for the HCF","Chapter 06","6 Long division",{"id":335,"type":43,"markdown":336},"division-prose","For big numbers, factorising can be hard. Try factorising 391 quickly! The **long division method** (also called **continued division**, or **Euclid’s algorithm**) avoids factorising altogether.\n\nThe rule:\n\n1. Divide the bigger number by the smaller one.\n2. If the remainder is 0, the divisor is the HCF. Stop.\n3. Otherwise, make the **old divisor** the new dividend and the **remainder** the new divisor. Go back to step 1.\n\nThe last non-zero remainder, which is the last divisor, is the HCF.\n\nWhy on earth should this work? Here is the short version: **any number that divides both 252 and 198 also divides their difference, and so it also divides the remainder** 252 − 198 = 54. So the pair (252, 198) has exactly the same common factors as the pair (198, 54), and the numbers keep shrinking until the answer is obvious. The Deepen layer turns this into a proper argument.",{"id":338,"type":129,"title":339,"problem":340,"steps":341,"help":347},"we-division","HCF of 252 and 198 by long division","Find HCF(252, 198) by the continued division method.",[342,343,344,345,346],"252 = 198 × 1 + 54. Remainder 54 is not 0, so divide again.","198 = 54 × 3 + 36. Remainder 36 is not 0, so divide again.","54 = 36 × 1 + 18. Remainder 18 is not 0, so divide again.","36 = 18 × 2 + 0. Remainder 0 is 0, so the HCF is the last divisor, **18**.","Check by factorising: 252 = 2² × 3² × 7 and 198 = 2 × 3² × 11. Shared: 2 × 3² = 18. ✓",{"simplerExplanation":348},"Keep dividing the last divisor by the last remainder. When the remainder becomes 0, the number you just divided by is the HCF.",{"id":350,"type":351,"title":352,"items":353},"steps-division-layout","steps","How the working looks on paper",[354,358,362,366,369,372],{"title":355,"tag":356,"text":357},"Set up","198 ) 252 ( 1","Write the divisor 198 outside and the dividend 252 inside. 198 goes into 252 once.",{"title":359,"tag":360,"text":361},"Subtract","252 − 198 = 54","Write 198 under 252 and subtract. The remainder is 54.",{"title":363,"tag":364,"text":365},"Shift","54 ) 198 ( 3","The remainder 54 becomes the new divisor; the old divisor 198 becomes the dividend. 54 × 3 = 162.",{"title":359,"tag":367,"text":368},"198 − 162 = 36","Remainder 36. Shift again: 36 ) 54 ( 1.",{"title":359,"tag":370,"text":371},"54 − 36 = 18","Remainder 18. Shift again: 18 ) 36 ( 2.",{"title":373,"tag":374,"text":375},"Finish","36 − 36 = 0","Remainder 0. The last divisor, **18**, is the HCF.",{"id":377,"type":158,"component":218,"componentVersion":5,"config":378,"objective":396,"textAlternative":397,"help":398},"lab-division",{"sets":379,"views":394,"challenge":170},[380,382,384,387,391],[169,381],198,[165,383],126,[385,386],391,544,[388,389,390],12,18,30,[392,260,393],16,40,[395],"division","Step through long division for the HCF and the common division ladder for the LCM, one row at a time.","This lab shows the division methods one row at a time.\n\n**Long division for the HCF:**\n- HCF(252, 198): 252 = 198 × 1 + 54; 198 = 54 × 3 + 36; 54 = 36 × 1 + 18; 36 = 18 × 2 + 0. HCF = **18**.\n- HCF(84, 126): 126 = 84 × 1 + 42; 84 = 42 × 2 + 0. HCF = **42**.\n- HCF(391, 544): 544 = 391 × 1 + 153; 391 = 153 × 2 + 85; 153 = 85 × 1 + 68; 85 = 68 × 1 + 17; 68 = 17 × 4 + 0. HCF = **17**.\n\n**Common division ladder for the LCM:**\n- LCM(12, 18, 30): ÷ 2 → 6, 9, 15; ÷ 3 → 2, 3, 5. LCM = 2 × 3 × 2 × 3 × 5 = **180**.\n- LCM(16, 24, 40): ÷ 2 → 8, 12, 20; ÷ 2 → 4, 6, 10; ÷ 2 → 2, 3, 5. LCM = 2 × 2 × 2 × 2 × 3 × 5 = **240**.\n\nNotice 391 and 544: neither is easy to factorise in your head, but three divisions show their HCF is 17 (391 = 17 × 23 and 544 = 2⁵ × 17).",{"simplerExplanation":399},"HCF: keep dividing by the remainder until nothing is left. LCM: keep dividing the whole row by a prime that goes into at least two of the numbers.",{"id":401,"type":255,"itemId":402,"prompt":403,"check":404,"hints":406,"feedback":409},"practice-division","hcf-and-lcm.understand-division-hcf","Use long division to find **HCF(391, 544)**.",{"kind":259,"answer":405,"tolerance":261},17,[407,408],"544 = 391 × 1 + 153. Now divide 391 by 153.","Keep going until the remainder is 0.",{"correct":410,"incorrect":411},"544 = 391 × 1 + 153; 391 = 153 × 2 + 85; 153 = 85 × 1 + 68; 85 = 68 × 1 + 17; 68 = 17 × 4 + 0. HCF = 17.","Steps: 544 = 391 × 1 + 153; 391 = 153 × 2 + 85; 153 = 85 × 1 + 68; 85 = 68 × 1 + 17; 68 = 17 × 4 + 0. The last divisor is 17.",{"id":413,"type":53,"title":414,"eyebrow":415,"navLabel":416},"ch07","Method 4: common division (the ladder) for the LCM","Chapter 07","7 Ladder method",{"id":418,"type":43,"markdown":419},"ladder-prose","The **common division method** finds the LCM of several numbers at once. Write the numbers in a row. Divide by a prime that divides **at least two** of them. Numbers it divides are replaced by the quotient; numbers it does not divide are **brought down** unchanged. Repeat until no prime divides two of the numbers in the row.\n\nThe LCM is the product of **all the divisors on the left** and **all the numbers left in the bottom row**.\n\nMany textbooks carry on dividing until every number in the bottom row is 1, even when a prime divides only one number. Both ways give the same LCM; the version above is just shorter.",{"id":421,"type":129,"title":422,"problem":423,"steps":424},"we-ladder","LCM of 12, 18 and 30 by common division","Find LCM(12, 18, 30) by the common division method.",[425,426,427,428,429,430],"Row: 12, 18, 30.","Divide by 2: row becomes 6, 9, 15.","Divide by 3: row becomes 2, 3, 5.","No prime divides two of 2, 3, 5 (they are co-prime in pairs). Stop.","LCM = 2 × 3 × 2 × 3 × 5 = **180**.","Check: 180 ÷ 12 = 15, 180 ÷ 18 = 10, 180 ÷ 30 = 6. ✓",{"id":432,"type":47,"variant":201,"title":433,"markdown":434},"careful-ladder-hcf","The ladder does not give the HCF of three numbers directly","In the ladder for 12, 18 and 30, the divisors on the left were 2 and 3, so it is tempting to say the HCF is 2 × 3 = 6. That happens to be right here, because both 2 and 3 divided **all three** numbers. But in general a divisor may have divided only **two** of the three.\n\nExample: 4, 6 and 9. The ladder divides by 2 (4 and 6 → 2, 3, 9) then by 3 (3 and 9 → 2, 1, 3). Divisors: 2 and 3, product 6. But HCF(4, 6, 9) = **1**, since no number except 1 divides all three. For the HCF, only count divisors that divide **every** number in the row.",{"id":436,"type":53,"title":437,"eyebrow":438,"navLabel":439},"ch08","HCF and LCM of three numbers","Chapter 08","8 Three numbers",{"id":441,"type":43,"markdown":442},"three-prose","Every method extends to three (or more) numbers.\n\n- **Prime factorisation:** for the HCF take the primes common to **all three**, with the smallest power; for the LCM take every prime that appears in **any** of them, with the highest power.\n- **Two at a time:** HCF(a, b, c) = HCF(HCF(a, b), c), and LCM(a, b, c) = LCM(LCM(a, b), c). So you can use long division twice.\n- **Ladder:** works directly for the LCM, as you just saw.",{"id":444,"type":129,"title":445,"problem":446,"steps":447},"we-three","HCF and LCM of 18, 30 and 42","Find HCF(18, 30, 42) and LCM(18, 30, 42).",[448,449,450,451,452,453],"18 = 2 × 3²; 30 = 2 × 3 × 5; 42 = 2 × 3 × 7.","Primes in all three: 2 (power 1 in each) and 3 (powers 2, 1, 1 → smallest 1).","HCF = 2 × 3 = **6**.","Primes in any: 2, 3, 5, 7. Highest powers: 2¹, 3², 5¹, 7¹.","LCM = 2 × 9 × 5 × 7 = **630**.","Two-at-a-time check: HCF(18, 30) = 6, then HCF(6, 42) = 6 ✓. LCM(18, 30) = 90, then LCM(90, 42) = 630 ✓.",{"id":455,"type":158,"component":218,"componentVersion":5,"config":456,"objective":473,"textAlternative":474,"help":475},"lab-three",{"sets":457,"views":471,"challenge":170},[458,460,461,464,468],[389,390,459],42,[388,389,390],[462,388,463],8,20,[465,466,467],4,6,9,[466,469,470],10,15,[472,229,395],"lists","Find the HCF and LCM of three numbers with lists, a three-circle Venn diagram and the ladder, and compare the methods.","This lab works with three numbers at once. In the Venn view, a prime in the middle of all three circles is shared by all three.\n\n- **18, 30, 42:** 18 = 2 × 3², 30 = 2 × 3 × 5, 42 = 2 × 3 × 7. HCF = **6**, LCM = **630**.\n- **12, 18, 30:** 12 = 2² × 3, 18 = 2 × 3², 30 = 2 × 3 × 5. HCF = **6**, LCM = **180**.\n- **8, 12, 20:** 8 = 2³, 12 = 2² × 3, 20 = 2² × 5. HCF = **4**, LCM = **120**.\n- **4, 6, 9:** 4 = 2², 6 = 2 × 3, 9 = 3². HCF = **1**, LCM = **36**.\n- **6, 10, 15:** 6 = 2 × 3, 10 = 2 × 5, 15 = 3 × 5. HCF = **1**, LCM = **30**.\n\nLook at 6, 10 and 15: every pair shares a factor (6 and 10 share 2, 6 and 15 share 3, 10 and 15 share 5), yet no number except 1 divides all three, so the HCF is 1. The LCM is 30.",{"simplerExplanation":476},"For the HCF, a prime must be in all three recipes. For the LCM, a prime only needs to be in one of them.",{"id":478,"type":255,"itemId":479,"prompt":480,"check":481,"hints":482,"feedback":485},"practice-three","hcf-and-lcm.understand-lcm-three","Find **LCM(8, 12, 20)**.",{"kind":259,"answer":167,"tolerance":261},[483,484],"8 = 2³, 12 = 2² × 3, 20 = 2² × 5.","Take the highest power of every prime that appears.",{"correct":486,"incorrect":487},"Yes: 2³ × 3 × 5 = 120.","Highest powers: 2³ (from 8), 3 (from 12), 5 (from 20). LCM = 8 × 3 × 5 = 120.",{"id":489,"type":53,"title":490,"eyebrow":491,"navLabel":492},"ch09","Three facts that save time","Chapter 09","9 Key facts",{"id":494,"type":43,"markdown":495},"facts-prose","**Fact 1: HCF × LCM = product of the two numbers.** For 12 and 18: HCF = 6, LCM = 36, and 6 × 36 = 216 = 12 × 18. For 36 and 48: 12 × 144 = 1728 = 36 × 48. The Venn picture shows why: the two circles together use every prime of both numbers once, with the overlap counted twice. HCF × LCM also counts the overlap twice (once in the HCF, once in the LCM).\n\n**This fact is only for two numbers.** For 18, 30 and 42: HCF × LCM = 6 × 630 = 3780, but 18 × 30 × 42 = 22,680. Not equal!\n\n**Fact 2: co-primes.** If HCF(a, b) = 1, then LCM(a, b) = a × b. (Put HCF = 1 into Fact 1.) Example: LCM(8, 15) = 120.\n\n**Fact 3: the HCF always divides the LCM.** The HCF divides a, and a divides the LCM, so the HCF divides the LCM. For 36 and 48: 144 ÷ 12 = 12.",{"id":497,"type":498,"items":499},"formulas-facts","formulas",[500,503,506,509,512,515],{"expression":501,"caption":502},"HCF(a, b) × LCM(a, b) = a × b","Two numbers only. For 12 and 18: 6 × 36 = 216 = 12 × 18.",{"expression":504,"caption":505},"LCM(a, b) = a × b ÷ HCF(a, b)","Find the HCF first (fast by long division), then the LCM for free.",{"expression":507,"caption":508},"HCF(a, b) = 1 ⇒ LCM(a, b) = a × b","Co-prime numbers: nothing shared, so the LCM is the full product.",{"expression":510,"caption":511},"HCF(a, b) | LCM(a, b)","The HCF always divides the LCM exactly.",{"expression":513,"caption":514},"HCF ≤ min(a, b) ≤ max(a, b) ≤ LCM","A quick sense check on any answer.",{"expression":516,"caption":517},"a | b ⇒ HCF = a, LCM = b","When one number divides the other, e.g. HCF(6, 24) = 6, LCM(6, 24) = 24.",{"id":519,"type":129,"title":520,"problem":521,"steps":522},"we-missing","Finding the missing number","The HCF of two numbers is **6** and their LCM is **180**. One number is **36**. Find the other.",[523,524,525,526],"Use HCF × LCM = product of the two numbers.","6 × 180 = 1080 = 36 × (other number).","Other number = 1080 ÷ 36 = **30**.","Check: HCF(36, 30) = 6 ✓ and LCM(36, 30) = 180 ✓.",{"id":528,"type":529,"prompt":530,"options":531,"explanation":544},"predict-impossible","prediction","Could two numbers have **HCF 8** and **LCM 60**?",[532,535,538,541],{"id":533,"label":534},"a","Yes, for example 8 and 60",{"id":536,"label":537},"b","Yes, but only 24 and 20",{"id":539,"label":540},"c","No, because 8 does not divide 60",{"id":542,"label":543},"d","No, because the LCM must be odd","**No.** The HCF always divides the LCM (Fact 3), and 60 ÷ 8 = 7 remainder 4. Check option a: HCF(8, 60) is actually 4, not 8. Examiners like to set traps like this. Always test \"does the HCF divide the LCM?\" before you start calculating.",{"id":546,"type":53,"title":547,"eyebrow":548,"navLabel":549},"ch10","HCF and LCM in fractions","Chapter 10","10 Fractions",{"id":551,"type":43,"markdown":552},"fractions-prose","This is where most people use HCF and LCM every week without noticing.\n\n**Simplifying a fraction uses the HCF.** To write 84⁄126 in its simplest form, divide top and bottom by the **largest** number that divides both, HCF(84, 126) = 42. That gives **2⁄3** in one step. Dividing by a smaller common factor, like 2, also works, but you would have to keep going.\n\n**Adding or subtracting fractions uses the LCM.** To add 5⁄12 + 7⁄18 you need a common denominator: a common multiple of 12 and 18. The **smallest** one, LCM(12, 18) = 36, keeps the numbers small. It is called the **lowest common denominator**.",{"id":554,"type":129,"title":555,"problem":556,"steps":557,"help":563},"we-add-fractions","Adding 5⁄12 and 7⁄18","Work out 5⁄12 + 7⁄18 using the LCM.",[558,559,560,561,562],"LCM(12, 18) = 36. (12 = 2² × 3, 18 = 2 × 3², so LCM = 2² × 3² = 36.)","5⁄12 = 15⁄36, because 36 ÷ 12 = 3.","7⁄18 = 14⁄36, because 36 ÷ 18 = 2.","15⁄36 + 14⁄36 = **29⁄36**.","HCF(29, 36) = 1, so 29⁄36 is already in simplest form.",{"anotherExample":564},"Using 12 × 18 = 216 as the denominator also works: 90⁄216 + 84⁄216 = 174⁄216. But then you must simplify by HCF(174, 216) = 6 to get 29⁄36. The LCM saves that last step.",{"id":566,"type":255,"itemId":567,"prompt":568,"check":569,"hints":573,"feedback":576},"practice-simplify","hcf-and-lcm.understand-simplify-fraction","Write **96⁄120** in its simplest form.",{"kind":570,"numerator":465,"denominator":571,"acceptEquivalent":572},"fraction",5,false,[574,575],"Find HCF(96, 120).","Divide top and bottom by it.",{"correct":577,"incorrect":578},"Yes: HCF(96, 120) = 24, so 96⁄120 = 4⁄5.","HCF(96, 120) = 24. 96 ÷ 24 = 4 and 120 ÷ 24 = 5, giving 4⁄5.",{"id":580,"type":255,"itemId":581,"prompt":582,"check":583,"hints":585,"feedback":588},"practice-add","hcf-and-lcm.understand-add-fractions","Work out **3⁄8 + 5⁄12** and give the answer in simplest form.",{"kind":570,"numerator":584,"denominator":260,"acceptEquivalent":572},19,[586,587],"LCM(8, 12) = ?","Convert both fractions to that denominator.",{"correct":589,"incorrect":590},"Yes: LCM(8, 12) = 24, 3⁄8 = 9⁄24, 5⁄12 = 10⁄24, total 19⁄24.","LCM(8, 12) = 24. 3⁄8 = 9⁄24 and 5⁄12 = 10⁄24. 9 + 10 = 19, so 19⁄24.",{"id":592,"type":53,"title":593,"eyebrow":594,"navLabel":595},"ch11","Common mix-ups and choosing the right one","Chapter 11","11 Mix-ups",{"id":597,"type":47,"variant":250,"title":598,"markdown":599},"misconception-add","\"The LCM of 4 and 6 is 4 + 6 = 10\"","Adding has nothing to do with HCF or LCM. 10 is not even a multiple of 4. Always check: does each number divide your LCM? Does your HCF divide each number?",{"id":601,"type":47,"variant":250,"title":602,"markdown":603},"misconception-hcf-prime","\"The HCF must be a prime number\"","The HCF is often composite. HCF(36, 48) = 12 and HCF(252, 198) = 18. It is the product of **all** the shared primes, not just one of them. Stopping at the first shared prime (2, for 36 and 48) gives a common factor, but not the highest one.",{"id":605,"type":47,"variant":250,"title":606,"markdown":607},"misconception-coprime-prime","\"Co-prime numbers must both be prime\"","Co-prime is about the **pair**, not about each number. 8 and 9 are both composite, but HCF(8, 9) = 1, so they are co-prime. Meanwhile 3 and 6 contains a prime, yet HCF(3, 6) = 3, so they are not co-prime. Two different primes, though, are always co-prime.",{"id":609,"type":47,"variant":610,"title":611,"markdown":612},"nuance-three-product","nuance","The product rule and three numbers","Learners often apply HCF × LCM = product to three numbers. It fails: for 2, 4 and 8, HCF × LCM = 2 × 8 = 16, but 2 × 4 × 8 = 64. Use it only for exactly two numbers. The Investigate and Deepen layers explore why.",{"id":614,"type":158,"component":615,"componentVersion":5,"config":616,"objective":644,"textAlternative":645},"lab-match-understand","match-pairs",{"prompt":617,"mode":618,"pairs":619},"Connect each pair of numbers to its HCF or LCM.","connect",[620,623,626,629,632,635,638,641],{"a":621,"b":622},"HCF(36, 48)","12",{"a":624,"b":625},"LCM(36, 48)","144",{"a":627,"b":628},"HCF(252, 198)","18",{"a":630,"b":631},"LCM(12, 18, 30)","180",{"a":633,"b":634},"HCF(28, 45)","1",{"a":636,"b":637},"LCM(8, 15)","120",{"a":639,"b":640},"HCF(18, 30, 42)","6",{"a":642,"b":643},"LCM(60, 84)","420","Match each HCF or LCM expression to its value, using whichever method you like.","This game shows eight expressions on one side and eight values on the other; connect each to its value.\n\n- HCF(36, 48) = **12** (2² × 3)\n- LCM(36, 48) = **144** (2⁴ × 3²)\n- HCF(252, 198) = **18** (by long division)\n- LCM(12, 18, 30) = **180** (by the ladder)\n- HCF(28, 45) = **1** (co-prime)\n- LCM(8, 15) = **120** (co-prime, so the product)\n- HCF(18, 30, 42) = **6**\n- LCM(60, 84) = **420**\n\nUse the sense check to eliminate: an HCF value must be no bigger than the smallest number in its brackets.",{"id":647,"type":158,"component":648,"componentVersion":5,"config":649,"objective":699,"textAlternative":700},"lab-sort-understand","sort-game",{"prompt":650,"bins":651,"items":658,"seconds":261},"Decide whether each word problem needs the HCF or the LCM.",[652,655],{"id":653,"label":654},"hcf","HCF",{"id":656,"label":657},"lcm","LCM",[659,663,667,671,675,679,683,687,691,695],{"id":660,"label":661,"bin":653,"why":662},"u1","Greatest length of a measuring rod that measures 4 m 50 cm and 6 m 30 cm exactly","The rod must divide 450 cm and 630 cm exactly; the greatest such length is HCF = 90 cm.",{"id":664,"label":665,"bin":656,"why":666},"u2","Least number of sweets that can be shared equally among 12, 15 or 20 children","The number must be divisible by 12, 15 and 20; the least is LCM = 60.",{"id":668,"label":669,"bin":656,"why":670},"u3","Three traffic lights change every 48 s, 72 s and 108 s. When do they change together?","Repeating cycles meet at LCM(48, 72, 108) = 432 s = 7 min 12 s.",{"id":672,"label":673,"bin":653,"why":674},"u4","Largest number of identical gift bags from 144 pens and 180 pencils","The number of bags must divide both amounts; the largest is HCF(144, 180) = 36.",{"id":676,"label":677,"bin":656,"why":678},"u5","Smallest square that can be made from 6 cm × 8 cm tiles","The square side must be a multiple of both 6 and 8: LCM = 24 cm.",{"id":680,"label":681,"bin":653,"why":682},"u6","Largest square tile for a hall 18 m 72 cm long and 13 m 20 cm wide","The tile side must divide 1,872 cm and 1,320 cm: HCF = 24 cm.",{"id":684,"label":685,"bin":656,"why":686},"u7","Three runners take 60 s, 72 s and 90 s per lap. When are they together at the start?","Laps repeat; they meet at LCM(60, 72, 90) = 360 s = 6 minutes.",{"id":688,"label":689,"bin":653,"why":690},"u8","Three tankers of 403 L, 434 L and 465 L: biggest can that measures each exactly","The can size must divide all three: HCF = 31 L.",{"id":692,"label":693,"bin":656,"why":694},"u9","Smallest number which leaves remainder 3 when divided by 4, 6 and 9","Subtract 3 and it is divisible by all three: LCM(4, 6, 9) + 3 = 36 + 3 = 39.",{"id":696,"label":697,"bin":653,"why":698},"u10","Largest number that divides 70 and 125 leaving remainders 5 and 8","It divides 70 − 5 = 65 and 125 − 8 = 117 exactly: HCF(65, 117) = 13.","Sort harder, textbook-style word problems into HCF and LCM, including measuring rods, smallest squares and remainder problems.","This game has ten word-problem cards to drop into an HCF bin or an LCM bin.\n\nHCF: the greatest measuring rod for 4 m 50 cm and 6 m 30 cm (90 cm); the most identical gift bags from 144 pens and 180 pencils (36); the largest square tile for a hall 18 m 72 cm by 13 m 20 cm (24 cm); the biggest can measuring 403 L, 434 L and 465 L (31 L); the largest number dividing 70 and 125 leaving remainders 5 and 8 (13, from HCF of 65 and 117).\n\nLCM: the least number of sweets shareable among 12, 15 or 20 children (60); traffic lights every 48, 72 and 108 s (together every 432 s); the smallest square from 6 cm × 8 cm tiles (side 24 cm); runners with 60, 72 and 90 s laps (360 s); the smallest number leaving remainder 3 when divided by 4, 6 and 9 (39).\n\nThe tricky ones: \"smallest square from tiles\" is LCM, because the square is built up from tiles, not cut into them.",{"id":702,"type":129,"title":703,"problem":704,"steps":705},"we-tank","The biggest can for three tankers","Three milk tankers hold **403 L**, **434 L** and **465 L**. What is the biggest can that can measure the milk of each tanker an exact number of times?",[706,707,708,709],"The can size must divide all three amounts exactly, and we want the biggest: this is an HCF problem.","Long division: 434 = 403 × 1 + 31; 403 = 31 × 13 + 0. So HCF(403, 434) = 31.","Now HCF(31, 465): 465 = 31 × 15 + 0. So the HCF of all three is **31 L**.","Check: 403 ÷ 31 = 13, 434 ÷ 31 = 14, 465 ÷ 31 = 15. ✓",{"id":711,"type":53,"title":712,"eyebrow":713,"navLabel":714},"ch12","One more of each method, and word problems","Chapter 12","12 Method practice",{"id":716,"type":129,"title":717,"problem":718,"steps":719},"we-listing-2","Listing: HCF and LCM of 16 and 40","Find HCF(16, 40) and LCM(16, 40) by listing.",[720,721,722,723],"Factors of 16: 1, 2, 4, 8, 16. Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40.","Common factors: 1, 2, 4, 8. **HCF = 8**.","Multiples of 40: 40, 80, 120. Test with 16: 40 ✗, **80** ✓ (16 × 5).","**LCM = 80**. Check with the product rule: 8 × 80 = 640 = 16 × 40 ✓.",{"id":725,"type":129,"title":726,"problem":727,"steps":728},"we-venn-2","Venn method: 120 and 144","Find HCF(120, 144) and LCM(120, 144) with a prime-factor Venn diagram.",[729,730,731,732,733],"120 = 2 × 2 × 2 × 3 × 5 = 2³ × 3 × 5. 144 = 2 × 2 × 2 × 2 × 3 × 3 = 2⁴ × 3².","Pair off shared primes: three 2s and one 3 are in both. **Overlap: 2 × 2 × 2 × 3**.","Only in 120: 5. Only in 144: 2 × 3.","HCF = product of the overlap = **24**.","LCM = product of everything = 5 × 24 × 2 × 3 = **720**.",{"id":735,"type":129,"title":736,"problem":737,"steps":738},"we-division-2","Long division with bigger numbers: 4,052 and 12,576","Find HCF(4,052, 12,576) by the long division method.",[739,740,741,742,743,744,745,746],"12,576 = 4,052 × 3 + 420.","4,052 = 420 × 9 + 272.","420 = 272 × 1 + 148.","272 = 148 × 1 + 124.","148 = 124 × 1 + 24.","124 = 24 × 5 + 4.","24 = 4 × 6 + 0.","The last divisor is **4**, so HCF(4,052, 12,576) = 4. Seven short divisions, and no factorising of five-digit numbers needed.",{"id":748,"type":129,"title":749,"problem":750,"steps":751},"we-ladder-2","Ladder with four numbers: 15, 25, 35 and 40","Find LCM(15, 25, 35, 40) by common division.",[752,753,754,755,756],"Row: 15, 25, 35, 40.","5 divides 15, 25, 35 and 40. Divide: row becomes 3, 5, 7, 8.","No prime divides two of 3, 5, 7, 8: 3, 5 and 7 are primes, 8 = 2³ shares nothing with them.","LCM = 5 × 3 × 5 × 7 × 8 = **4,200**.","Check: 4,200 ÷ 15 = 280, ÷ 25 = 168, ÷ 35 = 120, ÷ 40 = 105. All exact ✓.",{"id":758,"type":47,"variant":250,"title":759,"markdown":760},"misconception-last-quotient","\"The HCF is the last quotient\"","In long division there are two numbers on each line, the quotient and the divisor, and students sometimes report the wrong one. For HCF(455, 42): 455 = 42 × 10 + 35; 42 = 35 × 1 + 7; 35 = 7 × **5** + 0. The last quotient is 5, but the HCF is the last **divisor**, **7**.\n\nCheck any answer by dividing: 455 ÷ 7 = 65 and 42 ÷ 7 = 6, both exact. 455 ÷ 5 = 91 works, but 42 ÷ 5 does not.",{"id":762,"type":47,"variant":250,"title":763,"markdown":764},"misconception-left-only","\"In the ladder, the LCM is the product of the numbers on the left\"","For 12, 18, 30, the divisors on the left are 2 and 3. Multiplying only those gives 6, and some students write LCM = 6, which is smaller than all three numbers! The bottom row (2, 3, 5) must be multiplied in too: 2 × 3 × 2 × 3 × 5 = 180.\n\nThe sense check catches it at once: an LCM can never be smaller than the numbers themselves.",{"id":766,"type":47,"variant":250,"title":767,"markdown":768},"misconception-hcf-lcm-swap","\"The tile problem needs the LCM, because the floor is big\"","A common exam slip is choosing by the **size** of the numbers or by a single word. A floor of 360 cm × 270 cm is big, but the tiles must fit **inside** it: that is cutting the floor into equal squares, so HCF(360, 270) = 90 cm. The smallest square **made from** 6 cm × 8 cm tiles is built **up** from tiles, so it needs LCM(6, 8) = 24 cm. Ask \"am I breaking something down, or building\u002Fwaiting up?\" rather than looking for key words.",{"id":770,"type":129,"title":771,"problem":772,"steps":773},"we-metro","Mumbai local trains and a metro connection","At a station in Mumbai, a fast local stops every **12 minutes**, a slow local every **8 minutes** and a metro feeder bus every **20 minutes** (made-up timings, not a real timetable). All three are there together at 7:00 a.m. When are all three next together?",[774,775,776,777],"Three repeating timetables meeting: **LCM** problem.","12 = 2² × 3, 8 = 2³, 20 = 2² × 5.","Highest powers: 2³ and 3 and 5. LCM = 8 × 3 × 5 = **120 minutes**.","120 minutes after 7:00 a.m. is **9:00 a.m.**",{"id":779,"type":129,"title":780,"problem":781,"steps":782},"we-sarees","Cutting fabric for school uniforms","A tailor in Surat has three bolts of cloth: **84 m**, **126 m** and **210 m**. He wants to cut all of them into pieces of the **same length**, as long as possible, with nothing wasted. How long is each piece and how many pieces are there?",[783,784,785,786],"Cutting into the longest equal pieces: **HCF** problem.","84 = 2² × 3 × 7, 126 = 2 × 3² × 7, 210 = 2 × 3 × 5 × 7.","Primes in all three: 2, 3 and 7 (each with smallest power 1). HCF = 2 × 3 × 7 = **42 m**.","Pieces: 84 ÷ 42 = 2, 126 ÷ 42 = 3, 210 ÷ 42 = 5, so **10 pieces**.",{"id":788,"type":255,"itemId":789,"prompt":790,"check":791,"hints":793,"feedback":796},"practice-long-div-2","hcf-and-lcm.understand-division-455-42","Use long division to find **HCF(455, 42)**.",{"kind":259,"answer":792,"tolerance":261},7,[794,795],"455 = 42 × 10 + 35.","Now divide 42 by 35, and so on. Report the last divisor.",{"correct":797,"incorrect":798},"455 = 42 × 10 + 35; 42 = 35 × 1 + 7; 35 = 7 × 5 + 0. HCF = 7.","Divide 42 by 35 (remainder 7), then 35 by 7 (remainder 0). The last divisor, 7, is the HCF, not the last quotient 5.",{"id":800,"type":255,"itemId":801,"prompt":802,"check":803,"hints":804,"feedback":807},"practice-ladder-2","hcf-and-lcm.understand-ladder-4-6-10","Use the ladder to find **LCM(4, 6, 10)**.",{"kind":259,"answer":164,"tolerance":261},[805,806],"Divide the row by 2: 2, 3, 5.","Multiply the divisor and the bottom row.",{"correct":808,"incorrect":809},"Yes: 2 × 2 × 3 × 5 = 60.","4, 6, 10 ÷ 2 → 2, 3, 5 (no prime divides two of these). LCM = 2 × 2 × 3 × 5 = 60.",{"id":811,"type":255,"itemId":812,"prompt":813,"check":814,"hints":816,"feedback":819},"practice-crates","hcf-and-lcm.understand-vegetable-crates","At a sabzi mandi in Nashik, a trader has **150 onions**, **225 potatoes** and **300 tomatoes**. She packs them into identical crates, each with the same mix and nothing left over. What is the **greatest** number of crates?",{"kind":259,"answer":225,"tolerance":261,"unit":815},"crates",[817,818],"The number of crates must divide all three amounts.","Find HCF(150, 225, 300).",{"correct":820,"incorrect":821},"Yes: HCF = 75. Each crate gets 2 onions, 3 potatoes and 4 tomatoes.","150 = 2 × 3 × 5², 225 = 3² × 5², 300 = 2² × 3 × 5². Common: 3 × 5² = 75 crates.",{"id":823,"type":255,"itemId":824,"prompt":825,"check":826,"hints":828,"feedback":831},"practice-add-2","hcf-and-lcm.understand-add-3-10-7-15","Work out **3⁄10 + 7⁄15** in simplest form.",{"kind":570,"numerator":827,"denominator":390,"acceptEquivalent":572},23,[829,830],"LCM(10, 15) = 30.","3⁄10 = 9⁄30 and 7⁄15 = 14⁄30.",{"correct":832,"incorrect":833},"Yes: 9⁄30 + 14⁄30 = 23⁄30, already in simplest form since HCF(23, 30) = 1.","Lowest common denominator LCM(10, 15) = 30. 3⁄10 = 9⁄30, 7⁄15 = 14⁄30. Sum 23⁄30.",{"id":835,"type":53,"title":836,"eyebrow":837,"navLabel":838},"ch13","Check yourself","Chapter 13","13 Wrap-up",{"id":840,"type":841,"title":842,"questions":843},"quiz-understand","quiz","Methods and facts",[844,857,867,880,893,906,919,929,942,953],{"itemId":845,"prompt":846,"options":847,"correct":533,"why":856},"hcf-and-lcm.understand-q-hcf-pf","If a = 2³ × 3² × 5 and b = 2² × 3⁴ × 7, what is HCF(a, b)?",[848,850,852,854],{"id":533,"label":849},"2² × 3²",{"id":536,"label":851},"2³ × 3⁴",{"id":539,"label":853},"2³ × 3⁴ × 5 × 7",{"id":542,"label":855},"2 × 3","Common primes are 2 and 3; take the smaller powers: 2² and 3². 5 and 7 are not shared.",{"itemId":858,"prompt":859,"options":860,"correct":536,"why":866},"hcf-and-lcm.understand-q-lcm-pf","For the same a and b, LCM(a, b) = ?",[861,862,863,865],{"id":533,"label":849},{"id":536,"label":853},{"id":539,"label":864},"2⁵ × 3⁶ × 5 × 7",{"id":542,"label":851},"All primes that appear (2, 3, 5, 7), each with its highest power: 2³, 3⁴, 5, 7.",{"itemId":868,"prompt":869,"options":870,"correct":536,"why":879},"hcf-and-lcm.understand-q-division","In the long division method for HCF(1,071, 462), the first step is 1,071 = 462 × 2 + 147. What is the next division?",[871,873,875,877],{"id":533,"label":872},"1,071 ÷ 147",{"id":536,"label":874},"462 ÷ 147",{"id":539,"label":876},"147 ÷ 2",{"id":542,"label":878},"462 ÷ 2","The old divisor (462) is divided by the remainder (147). Carrying on: 462 = 147 × 3 + 21, then 147 = 21 × 7 + 0, so the HCF is 21.",{"itemId":881,"prompt":882,"options":883,"correct":536,"why":892},"hcf-and-lcm.understand-q-product","Two numbers have HCF 4 and LCM 96. One is 12. What is the other?",[884,886,888,890],{"id":533,"label":885},"8",{"id":536,"label":887},"32",{"id":539,"label":889},"24",{"id":542,"label":891},"48","4 × 96 = 384 = 12 × other, so other = 32. Check: HCF(12, 32) = 4, LCM(12, 32) = 96.",{"itemId":894,"prompt":895,"options":896,"correct":536,"why":905},"hcf-and-lcm.understand-q-coprime","Which pair is co-prime?",[897,899,901,903],{"id":533,"label":898},"14 and 21",{"id":536,"label":900},"15 and 28",{"id":539,"label":902},"12 and 27",{"id":542,"label":904},"22 and 55","15 = 3 × 5 and 28 = 2² × 7 share no prime, so HCF = 1. The others share 7, 3 and 11.",{"itemId":907,"prompt":908,"options":909,"correct":539,"why":918},"hcf-and-lcm.understand-q-impossible","Which could **not** be the HCF and LCM of two numbers?",[910,912,914,916],{"id":533,"label":911},"HCF 6, LCM 72",{"id":536,"label":913},"HCF 5, LCM 100",{"id":539,"label":915},"HCF 9, LCM 120",{"id":542,"label":917},"HCF 1, LCM 35","The HCF must divide the LCM. 120 ÷ 9 = 13 remainder 3, so HCF 9 and LCM 120 is impossible.",{"itemId":920,"prompt":921,"options":922,"correct":536,"why":928},"hcf-and-lcm.understand-q-three","HCF(24, 36, 60) = ?",[923,924,925,926],{"id":533,"label":640},{"id":536,"label":622},{"id":539,"label":889},{"id":542,"label":927},"360","24 = 2³ × 3, 36 = 2² × 3², 60 = 2² × 3 × 5. Common to all: 2² × 3 = 12.",{"itemId":930,"prompt":931,"options":932,"correct":539,"why":941},"hcf-and-lcm.understand-q-fraction","To add 7⁄15 + 3⁄20 with the smallest common denominator, which denominator do you use?",[933,935,937,939],{"id":533,"label":934},"5",{"id":536,"label":936},"35",{"id":539,"label":938},"60",{"id":542,"label":940},"300","The lowest common denominator is LCM(15, 20) = 60. 300 works too but is not the smallest.",{"itemId":943,"prompt":944,"options":945,"correct":539,"why":952},"hcf-and-lcm.understand-q-simplify","Which number should you divide by to simplify 60⁄84 in one step?",[946,948,950,951],{"id":533,"label":947},"2",{"id":536,"label":949},"4",{"id":539,"label":622},{"id":542,"label":643},"HCF(60, 84) = 12, giving 5⁄7 straight away.",{"itemId":954,"prompt":955,"options":956,"correct":536,"why":965},"hcf-and-lcm.understand-q-three-product","For 2, 3 and 4, is HCF × LCM equal to 2 × 3 × 4?",[957,959,961,963],{"id":533,"label":958},"Yes, it always is",{"id":536,"label":960},"No: 1 × 12 = 12, but 2 × 3 × 4 = 24",{"id":539,"label":962},"No: HCF × LCM is bigger",{"id":542,"label":964},"Yes, because they are small","HCF(2, 3, 4) = 1 and LCM(2, 3, 4) = 12; 1 × 12 = 12 ≠ 24. The product rule is only for two numbers.",{"id":967,"type":968,"title":969,"points":970},"cheat-sheet-understand","summary","Cheat sheet",[971,972,973,974,975,976,977,978,979,980,981],"**HCF (GCD, GCF):** largest number dividing each number exactly. **LCM:** smallest positive number each number divides.","**Listing:** fine for small numbers. For the LCM, skip-count the bigger number and test with the smaller.","**Prime factorisation:** HCF = common primes with **smallest** powers; LCM = all primes with **highest** powers.","**Venn picture:** shared primes in the overlap. HCF = product of the overlap; LCM = product of everything.","**Long division (Euclid):** divide, then divide the divisor by the remainder; the last non-zero remainder is the HCF.","**Ladder (common division):** divide the row by a prime that divides at least two numbers; LCM = divisors × bottom row.","**Three numbers:** HCF(a, b, c) = HCF(HCF(a, b), c); LCM likewise. Prime rules: HCF needs primes in all, LCM in any.","**Two numbers only:** HCF × LCM = a × b. Not true for three numbers.","**Co-primes:** HCF = 1, LCM = a × b. Co-prime numbers need not be prime (8 and 9).","**HCF always divides LCM.** Use this to spot impossible questions.","**Fractions:** HCF to simplify in one step; LCM for the lowest common denominator when adding or subtracting.",{"id":983,"type":984,"prompt":985},"reflect-understand","reflection","Which of the four methods would you choose for (a) HCF(18, 24), (b) HCF(1,147, 1,591), (c) LCM(6, 10, 14, 21)? Explain why each choice is the quickest, and what could go wrong with each method.",{"id":987,"type":988,"conceptId":989,"relation":990,"explanation":991},"conn-prime-u","connection","prime-and-composite","helps_understand","Prime factorisation and factor trees are the engine of the prime-factor method; co-primes have HCF 1.",{"id":993,"type":988,"conceptId":994,"relation":995,"explanation":996},"conn-four-ops-u","four-operations","applied_in","Long division with remainders is the heart of Euclid’s method for the HCF.",{"id":998,"type":988,"conceptId":999,"relation":1000,"explanation":1001},"conn-properties-u","properties-of-numbers","related_to","Divisibility rules and properties of multiplication explain why the HCF divides every common factor.",{"id":1003,"type":1004,"sourceIds":1005},"sources-understand","sources",[1006,1007,1008,1009,1010,1011,1012],"hcf-and-lcm-ncert-class6-playing-with-numbers","hcf-and-lcm-ncert-class10-real-numbers","hcf-and-lcm-mathsisfun-gcf","hcf-and-lcm-mathsisfun-lcm","hcf-and-lcm-khan-factors-multiples","hcf-and-lcm-wiki-gcd","hcf-and-lcm-nrich-factors-multiples-primes",[1006,1007,1008,1009,1010,1011,1012],"needs_review",{"generatedBy":1016,"notes":1017},"claude-code","Draft generated with Python-checked arithmetic; pending owner review.","7a936d327b6acb06118ec8733d1e65607cd7c4485a3864a9876495c748a960c4",{"component:factor-tree@1":1020,"component:hcf-lcm@1":1021,"logic:practice":1022,"component:match-pairs@1":1023,"component:sort-game@1":1024,"source:hcf-and-lcm-khan-factors-multiples":1025,"source:hcf-and-lcm-mathsisfun-gcf":1026,"source:hcf-and-lcm-mathsisfun-lcm":1027,"source:hcf-and-lcm-ncert-class10-real-numbers":1028,"source:hcf-and-lcm-ncert-class6-playing-with-numbers":1029,"source:hcf-and-lcm-nrich-factors-multiples-primes":1030,"source:hcf-and-lcm-wiki-gcd":1031},"dfb598a254f20e70550d586121be496f7a546ba0254060fd97b95cb919dfd934","83f068508b17b70184e45fbcf95d356790bfa2b0a98dcdb859f8e9dc105e5d33","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","01af9d960126bcdac368f15ef1b43edbf2e389c0ce966c7e903721b346655feb","8499aa4d85d927e518cacd55fa1f7870d1da1332351d07b641c2a1cc6c885659","722e7f7db3c09e9e29bf5c0f363173f52eb56ef08d1bd84ad3f51890d1375d07","48d1fd4c018916e9edeb2f76b15a8eb3cb83c72b08a9d3ab332a3c167fe9be0b","82c5fcc908046567f1a1bdb191f9b28fd42600389720d981bc1965d0c1542772","25674a9fbe152fe3c9ce7273cc44537dd3ebb9b393ed8ba01f12daf10e9d473f","857640343ebbf479db694438bf4af134fd5baee47f017b64110ded8b40bb2dc6",{"state":1033,"reviewer":1034,"selfReview":170,"reviewedAt":1035,"method":1036},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899598523]