[{"data":1,"prerenderedAt":1229},["ShallowReactive",2],{"layer:hcf-and-lcm:investigate":3},{"layer":4,"contentHash":1211,"dependencyHashes":1212,"approval":1223,"releaseId":1228},{"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":1206,"reviewStatus":1207,"authoring":1208},1,"hcf-and-lcm","en","investigate","Predict, test and explain: HCF and LCM patterns","Always, sometimes or never? Find out with your own experiments","Make predictions and test them: when the LCM equals the product, why neighbours are co-prime, how HCF × LCM = a × b holds for two numbers but not three, what scaling does, how remainder puzzles work, and how changing a word problem changes the answer.",[13,14,15,16,17],"Use predictions, tables and counterexamples to test statements about HCF and LCM.","Discover and explain that HCF × LCM = a × b for two numbers, and find when it fails for three.","Explain why the HCF of two numbers divides their difference, and use it for consecutive numbers.","Solve remainder problems by shifting to an HCF or LCM problem.","Predict how HCF and LCM change when the numbers in a problem change.",45,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Investigate",{"label":26,"value":27},"Reading time","≈ 45 minutes",{"label":29,"value":30},"Prior knowledge","The four methods (Understand)",{"label":32,"value":33},"Chapters","12",{"label":35,"value":36},"Predictions","9 to commit to",{"label":38,"value":39},"Labs","Frogs, neighbours, Venn, sort, match",[41,45,78,84,117,123,141,170,219,224,238,243,256,287,290,295,308,335,359,364,382,387,390,439,464,468,473,484,515,518,533,538,551,572,577,588,593,596,600,611,622,654,666,671,674,685,690,695,698,711,739,743,809,814,826,856,867,901,905,910,913,932,944,954,967,972,975,988,1014,1018,1030,1035,1065,1070,1162,1177,1181,1187,1191,1196],{"id":42,"type":43,"markdown":44},"intro-investigate","prose","In this layer **you** are the mathematician. Instead of being told a rule, you will make a guess, test it on lots of examples, look for a pattern, and then ask the most important question in mathematics: **is it always true, or only sometimes?**\n\nEvery chapter starts with a question and a prediction. Commit to an answer before you read on. Being wrong is part of the method: a surprising result is the best clue that there is something to explain.",{"id":46,"type":47,"title":48,"items":49},"steps-method","steps","How mathematicians investigate",[50,54,58,62,66,70,74],{"title":51,"tag":52,"text":53},"Ask","a clear question","For example: when is the LCM of two numbers equal to their product?",{"title":55,"tag":56,"text":57},"Predict","commit first","Write down what you think will happen, and why.",{"title":59,"tag":60,"text":61},"Test","many cases","Try small numbers, big numbers, odd and even numbers, primes and non-primes.",{"title":63,"tag":64,"text":65},"Tabulate","organise","Put results in a table so patterns are easy to see.",{"title":67,"tag":68,"text":69},"Conjecture","state a rule","A conjecture is a guess that fits all your evidence so far.",{"title":71,"tag":72,"text":73},"Hunt for counterexamples","try to break it","One example that breaks a rule is enough to prove the rule false.",{"title":75,"tag":76,"text":77},"Explain","why?","A rule you can explain is much stronger than a rule you have only tested.",{"id":79,"type":80,"variant":81,"title":82,"markdown":83},"def-counterexample","callout","definition","Conjecture and counterexample","A **conjecture** is a statement you believe is true because it fits your evidence, but have not yet proved.\n\nA **counterexample** is a single case that makes a statement false. \"All co-prime numbers are prime\" is false because of the counterexample 8 and 9. Checking a million examples cannot prove a rule, but one counterexample can destroy it.",{"id":85,"type":86,"title":87,"terms":88},"glossary-investigate","glossary","Investigation words",[89,93,97,101,105,109,113],{"term":90,"meaning":91,"example":92},"conjecture","A statement you think is true because it fits the evidence, but have not yet proved.","Conjecture: HCF × LCM = a × b for any two numbers.",{"term":94,"meaning":95,"example":96},"counterexample","One example that shows a general statement is false.","4 and 6 is a counterexample to \"LCM is always the product\".",{"term":98,"meaning":99,"example":100},"consecutive numbers","Whole numbers that follow one another with a gap of 1.","35 and 36",{"term":102,"meaning":103,"example":104},"pairwise co-prime","A group of numbers in which every pair has HCF 1.","3, 4, 5 are pairwise co-prime; 6, 10, 15 are not.",{"term":106,"meaning":107,"example":108},"always \u002F sometimes \u002F never true","The three possible verdicts on a statement: true for every case, true for some cases only, or true for no case.","LCM = a × b is sometimes true.",{"term":110,"meaning":111,"example":112},"scale (a pair)","Multiply both numbers by the same whole number.","Scaling 4 and 6 by 10 gives 40 and 60.",{"term":114,"meaning":115,"example":116},"remainder","What is left after dividing as many whole times as possible.","62 ÷ 5 = 12 remainder 2",{"id":118,"type":119,"title":120,"eyebrow":121,"navLabel":122},"ch01","chapter","When is the LCM equal to the product?","Chapter 01","1 LCM vs product",{"id":124,"type":125,"prompt":126,"options":127,"explanation":140},"predict-lcm-product","prediction","For which of these pairs is the LCM **equal to** the product of the two numbers?",[128,131,134,137],{"id":129,"label":130},"a","4 and 6",{"id":132,"label":133},"b","5 and 7",{"id":135,"label":136},"c","8 and 9",{"id":138,"label":139},"d","Both 5 and 7, and 8 and 9","**Both 5 and 7, and 8 and 9.** LCM(5, 7) = 35 = 5 × 7, and LCM(8, 9) = 72 = 8 × 9. But LCM(4, 6) = 12, only half of 4 × 6 = 24. The frogs lab below lets you test many more pairs. Look for what the pairs with LCM = product have in common.",{"id":142,"type":143,"component":144,"componentVersion":5,"config":145,"objective":164,"textAlternative":165,"help":166},"lab-frogs-inv","interactive","hcf-lcm",{"sets":146,"views":161,"challenge":163},[147,150,152,155,157,160],[148,149],3,4,[149,151],6,[153,154],5,7,[151,156],9,[158,159],8,12,[158,156],[162],"frogs",true,"Race two frogs with different jump sizes, predict the first shared stone, and compare it with the product of the jumps.","Two frogs start at 0 and jump along numbered stones, each with a fixed jump size. The first stone they share is the LCM. For each pair, compare it with the product of the jumps.\n\n- Jumps **3 and 4:** first shared stone 12; product 12; **equal**. HCF = 1.\n- Jumps **4 and 6:** first shared stone 12; product 24; product is 2 times the LCM. HCF = 2.\n- Jumps **5 and 7:** first shared stone 35; product 35; **equal**. HCF = 1.\n- Jumps **6 and 9:** first shared stone 18; product 54; product is 3 times the LCM. HCF = 3.\n- Jumps **8 and 12:** first shared stone 24; product 96; product is 4 times the LCM. HCF = 4.\n- Jumps **8 and 9:** first shared stone 72; product 72; **equal**. HCF = 1.\n\nPattern: the LCM equals the product exactly when the HCF is 1. Otherwise the product is HCF times too big.",{"simplerExplanation":167,"hints":168},"When the two jumps share a factor, the frogs meet early. When they share nothing but 1, they have to wait until the product.",[169],"Look at the HCF column in your results. What is special about the pairs where LCM = product?",{"id":171,"type":172,"caption":173,"columns":174,"rows":180},"table-lcm-product","table","Testing ten pairs (all computed): what is the product divided by the LCM?",[175,176,177,178,179],"Pair","Product a × b","LCM","Product ÷ LCM","HCF",[181,184,188,191,196,200,203,207,212,215],[182,33,33,183,183],"3, 4","1",[185,186,33,187,187],"4, 6","24","2",[189,190,190,183,183],"5, 7","35",[192,193,194,195,195],"6, 9","54","18","3",[197,198,186,199,199],"8, 12","96","4",[201,202,202,183,183],"7, 10","70",[204,205,206,195,195],"9, 12","108","36",[208,209,210,211,211],"10, 15","150","30","5",[213,214,214,183,183],"8, 9","72",[216,217,206,218,218],"12, 18","216","6",{"id":220,"type":80,"variant":221,"title":222,"markdown":223},"aha-product-lcm","aha","The product is too big by exactly the HCF","Look at the last two columns: **product ÷ LCM is always the HCF**. So a × b = HCF × LCM. When the HCF is 1 (co-primes), the LCM is the whole product. When the numbers share a factor, the frogs meet early, and they meet early by exactly the shared amount.\n\nWhy? The product a × b counts every shared prime twice (once from a, once from b). The LCM needs it only once. The HCF is exactly the collection of primes that got counted twice.",{"id":225,"type":226,"itemId":227,"prompt":228,"check":229,"hints":233,"feedback":235},"practice-product-hcf","practice","hcf-and-lcm.investigate-product-div","Without finding the LCM directly: the product of 18 and 24 is 432 and HCF(18, 24) = 6. What is LCM(18, 24)?",{"kind":230,"answer":231,"tolerance":232},"number",72,0,[234],"Product ÷ HCF = LCM.",{"correct":236,"incorrect":237},"Yes: 432 ÷ 6 = 72.","LCM = product ÷ HCF = 432 ÷ 6 = 72. Check: 72 ÷ 18 = 4 and 72 ÷ 24 = 3.",{"id":239,"type":119,"title":240,"eyebrow":241,"navLabel":242},"ch02","When is the HCF one of the numbers?","Chapter 02","2 HCF = a number",{"id":244,"type":125,"prompt":245,"options":246,"explanation":255},"predict-hcf-smaller","HCF(7, 28) = ? and LCM(7, 28) = ?",[247,249,251,253],{"id":129,"label":248},"HCF 1, LCM 196",{"id":132,"label":250},"HCF 7, LCM 28",{"id":135,"label":252},"HCF 7, LCM 196",{"id":138,"label":254},"HCF 4, LCM 28","**HCF 7, LCM 28.** 7 divides 28 (28 = 7 × 4), so 7 is a factor of both and nothing bigger than 7 can divide 7. So HCF = 7. And 28 is already a multiple of 7, so the first common multiple is 28 itself. Computed: HCF(7, 28) = 7, LCM(7, 28) = 28.",{"id":257,"type":172,"caption":258,"columns":259,"rows":261},"table-divides","When one number divides the other (all computed)",[175,260,179,177],"Does the smaller divide the larger?",[262,267,269,271,275,279,281,285],[263,264,265,266],"7, 28","yes","7","28",[268,264,218,210],"6, 30",[270,264,33,206],"12, 36",[272,264,273,274],"15, 45","15","45",[276,277,199,278],"8, 20","no","40",[280,277,195,214],"9, 24",[282,264,283,284],"25, 100","25","100",[286,277,265,202],"14, 35",{"id":288,"type":43,"markdown":289},"divides-prose","The table shows a clean rule: **if a divides b, then HCF(a, b) = a and LCM(a, b) = b.** When the smaller number does not divide the larger, the HCF is smaller than both, and the LCM is bigger than both.\n\nThis gives a quick test for tricky questions. \"The HCF of two numbers is 15 and one of them is 15. What can you say?\" The other number must be a multiple of 15, and the LCM is that other number.",{"id":291,"type":119,"title":292,"eyebrow":293,"navLabel":294},"ch03","Neighbours: numbers next to each other","Chapter 03","3 Neighbours",{"id":296,"type":125,"prompt":297,"options":298,"explanation":307},"predict-consecutive","Pick any two **consecutive** whole numbers, like 20 and 21, or 35 and 36. What do you think their HCF is?",[299,301,303,305],{"id":129,"label":300},"It depends on the numbers",{"id":132,"label":302},"Always 1",{"id":135,"label":304},"Always 2",{"id":138,"label":306},"Always the smaller number","**Always 1.** Test: HCF(20, 21) = 1, HCF(35, 36) = 1, HCF(99, 100) = 1, HCF(143, 144) = 1. Here is the reason: any number that divides both n and n + 1 must also divide their difference, which is 1. The only whole number that divides 1 is 1. So neighbours are always co-prime, and their LCM is always their product.",{"id":309,"type":143,"component":144,"componentVersion":5,"config":310,"objective":331,"textAlternative":332,"help":333},"lab-neighbours",{"sets":311,"views":328,"challenge":163},[312,315,318,321,323,325],[313,314],20,21,[316,317],35,36,[319,320],14,16,[314,322],27,[324,316],30,[326,327],44,52,[329,330],"lists","venn","Test pairs of numbers that are close together and find how their difference limits their HCF.","This lab lists the factors of close pairs of numbers and shows their shared prime factors.\n\n- **20 and 21** (difference 1): HCF = **1**, LCM = **420**. The HCF 1 divides the difference 1.\n- **35 and 36** (difference 1): HCF = **1**, LCM = **1260**. The HCF 1 divides the difference 1.\n- **14 and 16** (difference 2): HCF = **2**, LCM = **112**. The HCF 2 divides the difference 2.\n- **21 and 27** (difference 6): HCF = **3**, LCM = **189**. The HCF 3 divides the difference 6.\n- **30 and 35** (difference 5): HCF = **5**, LCM = **210**. The HCF 5 divides the difference 5.\n- **44 and 52** (difference 8): HCF = **4**, LCM = **572**. The HCF 4 divides the difference 8.\n\nPattern: the HCF of two numbers always divides their difference. Numbers 1 apart have HCF 1; numbers 2 apart have HCF 1 or 2; numbers 6 apart have HCF 1, 2, 3 or 6.",{"simplerExplanation":334},"If a number fits exactly into both numbers, it also fits exactly into the gap between them.",{"id":336,"type":172,"caption":337,"columns":338,"rows":341},"table-difference","The HCF divides the difference (all computed)",[175,339,179,340],"Difference","Does the HCF divide the difference?",[342,344,346,348,350,352,354,357],[343,183,183,264],"20, 21",[345,187,183,264],"13, 15",[347,187,187,264],"14, 16",[349,218,195,264],"21, 27",[351,218,187,264],"22, 28",[353,218,183,264],"25, 31",[355,356,356,264],"30, 40","10",[358,273,273,264],"45, 60",{"id":360,"type":80,"variant":361,"title":362,"markdown":363},"observation-odd-even","observation","Consecutive odd and consecutive even numbers","- Two **consecutive odd** numbers (like 15 and 17) differ by 2, so their HCF divides 2. Both are odd, so it cannot be 2. It is **always 1**: HCF(15, 17) = 1.\n- Two **consecutive even** numbers (like 14 and 16) differ by 2 and are both even, so their HCF is **always 2**: HCF(14, 16) = 2, HCF(98, 100) = 2.",{"id":365,"type":226,"itemId":366,"prompt":367,"check":368,"hints":377,"feedback":379},"practice-diff","hcf-and-lcm.investigate-difference-hcf","Two numbers differ by **9**. Which of these could **not** be their HCF?",{"kind":369,"options":370,"correct":376},"choice",[371,372,373,374],{"id":129,"label":183},{"id":132,"label":195},{"id":135,"label":218},{"id":138,"label":375},"9",[135],[378],"The HCF must divide the difference.",{"correct":380,"incorrect":381},"6 does not divide 9, so it cannot be the HCF. 1, 3 and 9 are all possible (e.g. 10 and 19; 3 and 12; 9 and 18).","The HCF must divide 9, so it can only be 1, 3 or 9. 6 is impossible.",{"id":383,"type":119,"title":384,"eyebrow":385,"navLabel":386},"ch04","Testing the product rule HCF × LCM = a × b","Chapter 04","4 Product rule",{"id":388,"type":43,"markdown":389},"product-test-prose","In the first chapter you noticed that the product divided by the LCM was always the HCF. That is the same as saying **HCF × LCM = a × b**. Before believing it, a good investigator tries to break it: big numbers, primes, equal numbers, numbers where one divides the other, numbers with lots of shared factors.",{"id":391,"type":172,"caption":392,"columns":393,"rows":398},"table-product-tests","Trying to break HCF × LCM = a × b (all computed)",[394,395,179,177,396,397],"a, b","Why chosen","HCF × LCM","a × b",[399,403,408,414,420,425,430,434],[400,401,183,402,402,402],"1, 17","one number is 1","17",[404,405,406,406,407,407],"13, 13","equal numbers","13","169",[409,410,411,412,413,413],"16, 64","one divides the other","16","64","1,024",[415,416,417,418,419,419],"84, 126","lots shared","42","252","10,584",[421,422,406,423,424,424],"91, 143","two-prime products","1,001","13,013",[426,427,210,428,429,429],"210, 330","larger numbers","2,310","69,300",[431,432,183,433,433,433],"256, 243","powers of different primes","62,208",[435,436,33,437,438,438],"360, 588","three shared primes","17,640","2,11,680",{"id":440,"type":143,"component":144,"componentVersion":5,"config":441,"objective":460,"textAlternative":461,"help":462},"lab-product-rule",{"sets":442,"views":458,"challenge":163},[443,446,449,452,455],[444,445],84,126,[447,448],91,143,[450,451],210,330,[453,454],360,588,[456,457],256,243,[329,330,459],"division","Find the HCF and LCM of larger pairs with any view, and check that HCF × LCM equals the product every time.","This lab offers three views of each pair: factor lists, a prime-factor Venn diagram and the division ladder. For each pair, multiply the HCF by the LCM and compare with the product.\n\n- **84 and 126:** 84 = 2² × 3 × 7, 126 = 2 × 3² × 7. HCF = 42, LCM = 252. HCF × LCM = 10,584 = 84 × 126.\n- **91 and 143:** 91 = 7 × 13, 143 = 11 × 13. HCF = 13, LCM = 1,001. HCF × LCM = 13,013 = 91 × 143.\n- **210 and 330:** 210 = 2 × 3 × 5 × 7, 330 = 2 × 3 × 5 × 11. HCF = 30, LCM = 2,310. HCF × LCM = 69,300 = 210 × 330.\n- **360 and 588:** 360 = 2³ × 3² × 5, 588 = 2² × 3 × 7². HCF = 12, LCM = 17,640. HCF × LCM = 2,11,680 = 360 × 588.\n- **256 and 243:** 256 = 2⁸, 243 = 3⁵. HCF = 1, LCM = 62,208. HCF × LCM = 62,208 = 256 × 243.\n\nIn the Venn picture: the product uses the overlap twice; so does HCF × LCM (the overlap is in both). That is why the rule never breaks for two numbers.",{"simplerExplanation":463},"However big the numbers, HCF × LCM always equals the two numbers multiplied together.",{"id":465,"type":80,"variant":221,"title":466,"markdown":467},"aha-hcf-divides-lcm","A second rule falls out: the HCF divides the LCM","In every row above, the LCM is a whole number of HCFs: for 84 and 126, 252 ÷ 42 = 6. In fact LCM ÷ HCF = (a ÷ HCF) × (b ÷ HCF), which for 84 and 126 is 2 × 3 = 6. This is the quickest way to spot impossible questions: if a question says the HCF is 12 and the LCM is 100, you know immediately something is wrong.",{"id":469,"type":119,"title":470,"eyebrow":471,"navLabel":472},"ch05","Does the product rule work for three numbers?","Chapter 05","5 Three numbers",{"id":474,"type":125,"prompt":475,"options":476,"explanation":483},"predict-three","For three numbers a, b, c, is HCF × LCM always equal to a × b × c?",[477,479,481],{"id":129,"label":478},"Yes, always",{"id":132,"label":480},"Never",{"id":135,"label":482},"Sometimes: it depends on the numbers","**Sometimes.** The table below has cases where it works and cases where it fails badly. Your job is to find what the \"works\" cases have in common.",{"id":485,"type":172,"caption":486,"columns":487,"rows":491},"table-three","HCF × LCM against the product for three numbers (all computed)",[488,179,177,396,489,490],"a, b, c","a × b × c","Equal?",[492,494,497,500,503,507,510,513],[493,183,210,210,210,264],"2, 3, 5",[495,183,496,496,496,264],"3, 4, 5","60",[498,183,499,499,499,264],"5, 7, 9","315",[501,183,33,33,186,502],"2, 3, 4","**no**",[504,187,496,505,506,502],"4, 6, 10","120","240",[508,187,509,411,412,502],"2, 4, 8","8",[511,183,210,210,512,502],"6, 10, 15","900",[514,183,206,206,217,502],"4, 6, 9",{"id":516,"type":43,"markdown":517},"three-prose","It works for (2, 3, 5), (3, 4, 5) and (5, 7, 9). In each of those, **every pair is co-prime**: no two of the numbers share a factor. It fails as soon as any two share a factor, even when the HCF of all three is 1. Look at (6, 10, 15): HCF = 1 and LCM = 30, but the product is 900. Each pair shares a prime (6 and 10 share 2, 6 and 15 share 3, 10 and 15 share 5), and each of those shared primes is counted twice in the product, but it is not in the HCF of all three, so nothing makes up for it.\n\nSo the honest statement is: **for three numbers, HCF × LCM = a × b × c only when the numbers are pairwise co-prime** (and then both sides equal a × b × c with HCF = 1). The Deepen layer shows a correct three-number formula.",{"id":519,"type":143,"component":144,"componentVersion":5,"config":520,"objective":531,"textAlternative":532},"lab-three-venn",{"sets":521,"views":529,"challenge":530},[522,525,526,528],[151,523,524],10,15,[149,151,523],[527,149,158],2,[148,149,153],[330],false,"Place the prime factors of three numbers in a three-circle Venn diagram to see which primes are shared by two numbers but not all three.","This lab places the prime factors of three numbers in three overlapping circles: the centre holds primes shared by all three; the three petal regions hold primes shared by exactly two.\n\n- **6, 10, 15:** 6 = 2 × 3, 10 = 2 × 5, 15 = 3 × 5. HCF (centre) = 1, LCM (everything) = 30, product = 900.\n- **4, 6, 10:** 4 = 2², 6 = 2 × 3, 10 = 2 × 5. HCF (centre) = 2, LCM (everything) = 60, product = 240.\n- **2, 4, 8:** 2 = 2, 4 = 2², 8 = 2³. HCF (centre) = 2, LCM (everything) = 8, product = 64.\n- **3, 4, 5:** 3 = 3, 4 = 2², 5 = 5. HCF (centre) = 1, LCM (everything) = 60, product = 60.\n\nFor 6, 10, 15 the centre is empty but each petal has a prime (2, 3, 5): those primes are counted twice in the product and only once in the LCM. For 3, 4, 5 there are no shared primes at all, and HCF × LCM = product.",{"id":534,"type":119,"title":535,"eyebrow":536,"navLabel":537},"ch06","What happens when you scale the numbers?","Chapter 06","6 Scaling",{"id":539,"type":125,"prompt":540,"options":541,"explanation":550},"predict-scale","HCF(4, 6) = 2 and LCM(4, 6) = 12. What are HCF(40, 60) and LCM(40, 60)?",[542,544,546,548],{"id":129,"label":543},"HCF 2, LCM 12",{"id":132,"label":545},"HCF 20, LCM 120",{"id":135,"label":547},"HCF 20, LCM 1,200",{"id":138,"label":549},"HCF 200, LCM 120","**HCF 20, LCM 120.** Multiplying both numbers by 10 multiplies both the HCF and the LCM by 10. Computed: HCF(40, 60) = 20, LCM(40, 60) = 120. In prime terms, you have added one 2 and one 5 to both recipes, so they go into the overlap, and so into both the HCF and the LCM.",{"id":552,"type":172,"caption":553,"columns":554,"rows":559},"table-scale","Scaling both numbers by k (all computed)",[555,556,179,177,557,558],"k","Numbers 4k, 6k","HCF ÷ k","LCM ÷ k",[560,561,562,563,565,568],[183,185,187,33,187,33],[187,197,199,186,187,33],[195,216,218,206,187,33],[211,564,356,496,187,33],"20, 30",[356,566,567,505,187,33],"40, 60","20",[283,569,570,571,187,33],"100, 150","50","300",{"id":573,"type":80,"variant":574,"title":575,"markdown":576},"nuance-scale-one","nuance","Scaling only one number is different","If you multiply only **one** number, anything can happen. HCF(4, 6) = 2, and HCF(4, 12) = 4 (it doubled), but HCF(4, 18) = 2 (no change) and HCF(4, 30) = 2 (no change). It depends on which primes you added and whether the other number already had them.",{"id":578,"type":226,"itemId":579,"prompt":580,"check":581,"hints":583,"feedback":585},"practice-scale","hcf-and-lcm.investigate-scale-hcf","HCF(9, 12) = 3. Use scaling to find **HCF(900, 1,200)** without factorising.",{"kind":230,"answer":582,"tolerance":232},300,[584],"900 = 9 × 100 and 1,200 = 12 × 100.",{"correct":586,"incorrect":587},"Yes: 3 × 100 = 300.","Both numbers were multiplied by 100, so the HCF is 3 × 100 = 300.",{"id":589,"type":119,"title":590,"eyebrow":591,"navLabel":592},"ch07","Remainder puzzles: shifting by a fixed amount","Chapter 07","7 Remainders",{"id":594,"type":43,"markdown":595},"remainder-prose","Some of the most famous textbook problems use the **same remainder** trick. Investigate this one:\n\n> Find the smallest number that leaves remainder **2** when divided by **3, 4 and 5**.\n\nTry listing numbers that leave remainder 2 when divided by 5: 7, 12, 17, 22, 27, 32, 37, 42, 47, 52, 57, **62**, … Now test each with 3 and 4. The first one that works is **62**.\n\nWhy 62? Take 2 away and you get **60**, which is divisible by 3, 4 and 5. In fact 60 = LCM(3, 4, 5). So the rule is: **smallest number = LCM + remainder**. And the next ones are 60 × 2 + 2 = 122, 60 × 3 + 2 = 182, and so on.",{"id":597,"type":80,"variant":574,"title":598,"markdown":599},"nuance-two-itself","Is 2 the real answer?","Strictly, 2 itself leaves remainder 2 when divided by 3, 4 or 5 (2 = 3 × 0 + 2). Textbooks mean the smallest number **bigger than the divisors**, or say \"other than 2\". Whenever you see this type of problem, the full list of answers is 2, 62, 122, 182, …: the remainder plus any multiple of the LCM.",{"id":601,"type":125,"prompt":602,"options":603,"explanation":610},"predict-divide-remainder","What is the **largest** number that divides **43** and **91** leaving remainder **7** in each case?",[604,605,606,608],{"id":129,"label":265},{"id":132,"label":33},{"id":135,"label":607},"14",{"id":138,"label":609},"84","**12.** Take away the remainder first: 43 − 7 = 36 and 91 − 7 = 84 must both be divisible exactly by the number. The largest such number is HCF(36, 84) = 12. Check: 43 = 12 × 3 + 7 ✓ and 91 = 12 × 7 + 7 ✓. Note the divisor must be bigger than the remainder, and 12 > 7, so this works.",{"id":612,"type":613,"title":614,"problem":615,"steps":616},"we-different-remainders","worked_example","Different remainders with a hidden pattern","Find the smallest number that leaves remainders **3, 4 and 5** when divided by **5, 6 and 7** respectively.",[617,618,619,620,621],"The remainders are different, so \"LCM + remainder\" does not work directly.","But look: 5 − 3 = 2, 6 − 4 = 2, 7 − 5 = 2. Each remainder is **2 short** of the divisor.","So if we **add 2** to the number, each division becomes exact. The number + 2 is a common multiple of 5, 6 and 7.","LCM(5, 6, 7) = 210. So the number = 210 − 2 = **208**.","Check: 208 = 5 × 41 + 3 ✓; 208 = 6 × 34 + 4 ✓; 208 = 7 × 29 + 5 ✓.",{"id":623,"type":172,"caption":624,"columns":625,"rows":630},"table-remainder-types","Four remainder patterns (all answers computed)",[626,627,628,629],"Problem","Trick","Uses","Answer",[631,636,640,645,649],[632,633,634,635],"Smallest number leaving remainder 2 when divided by 3, 4, 5 (other than 2)","Subtract the same remainder","LCM + 2","62",[637,638,639,33],"Largest number dividing 43 and 91 leaving remainder 7","Subtract the remainder from each","HCF(36, 84)",[641,642,643,644],"Smallest number leaving remainders 3, 4, 5 when divided by 5, 6, 7","Same shortfall 2","LCM − 2","208",[646,647,648,406],"Largest number dividing 70 and 125 leaving remainders 5 and 8","Subtract each remainder","HCF(65, 117)",[650,651,652,653],"Smallest 4-digit number divisible by 12, 15 and 20","Multiples of LCM 60","Next multiple of 60 after 999","1,020",{"id":655,"type":226,"itemId":656,"prompt":657,"check":658,"hints":660,"feedback":663},"practice-remainder","hcf-and-lcm.investigate-remainder-1","What is the smallest number greater than 1 that leaves remainder **1** when divided by **2, 3, 4, 5 and 6**?",{"kind":230,"answer":659,"tolerance":232},61,[661,662],"Subtract 1: the result is divisible by 2, 3, 4, 5 and 6.","LCM(2, 3, 4, 5, 6) = ?",{"correct":664,"incorrect":665},"Yes: LCM = 60, so the number is 61.","The number − 1 must be a common multiple of 2, 3, 4, 5, 6. LCM = 60, so the number is 61.",{"id":667,"type":119,"title":668,"eyebrow":669,"navLabel":670},"ch08","Counting common multiples","Chapter 08","8 Counting",{"id":672,"type":43,"markdown":673},"counting-prose","How many numbers from 1 to 100 are divisible by **both 4 and 6**?\n\nYour first guess might be to count multiples of 4 (there are 25) and multiples of 6 (there are 16) and do something with them. But a number divisible by both 4 and 6 is a **common multiple**, and every common multiple is a multiple of the LCM, which is 12. So the question is really \"how many multiples of 12 are there up to 100?\" The answer is 100 ÷ 12 = 8 remainder 4, so **8**: 12, 24, 36, 48, 60, 72, 84, 96.\n\nA common wrong answer is to count multiples of 4 × 6 = 24: that gives only 4, and misses 12, 36, 60 and 84.",{"id":675,"type":125,"prompt":676,"options":677,"explanation":684},"predict-count","How many numbers from **1 to 200** are divisible by **both 6 and 8**?",[678,679,680,682],{"id":129,"label":199},{"id":132,"label":509},{"id":135,"label":681},"33",{"id":138,"label":683},"58","**8.** Divisible by both 6 and 8 means divisible by LCM(6, 8) = 24. 200 ÷ 24 = 8 remainder 8, so there are 8: 24, 48, 72, 96, 120, 144, 168, 192. Using 6 × 8 = 48 would give only 4, which is wrong.",{"id":686,"type":80,"variant":687,"title":688,"markdown":689},"example-calendar","example","A calendar investigation","A school library opens a special reading corner every **4th** day of the year and a story session every **6th** day (days 4, 8, 12, … and 6, 12, 18, …). On how many days in a 365-day year do both happen?\n\nBoth happen on days that are multiples of LCM(4, 6) = 12. 365 ÷ 12 = 30 remainder 5, so on **30 days**.",{"id":691,"type":119,"title":692,"eyebrow":693,"navLabel":694},"ch09","Word-problem detective: what if the numbers change?","Chapter 09","9 What if…?",{"id":696,"type":43,"markdown":697},"whatif-prose","Investigating is not just for pure numbers. Take the courtyard from Discover, 240 cm × 180 cm, which needed 60 cm tiles. What if the courtyard changes?\n\n- Make it **250 cm × 180 cm** (10 cm longer). Now HCF(250, 180) = **10 cm**. A tiny change made the best tile six times smaller, and you would need 450 tiles instead of 12.\n- Make it **300 cm × 180 cm**. HCF(300, 180) = **60 cm**, back to large tiles: 15 tiles.\n\nThis is why tilers and carpenters love \"round\" measurements like 240, 300 and 360: they have lots of factors, so many tile sizes fit.",{"id":699,"type":125,"prompt":700,"options":701,"explanation":710},"predict-bus-change","Buses A and B leave together every 60 minutes (A every 15 min, B every 20 min). The company changes bus B to every **25 minutes**. How often do they now leave together?",[702,704,706,708],{"id":129,"label":703},"Every 40 minutes",{"id":132,"label":705},"Every 60 minutes",{"id":135,"label":707},"Every 75 minutes",{"id":138,"label":709},"Every 375 minutes","**Every 75 minutes.** LCM(15, 25) = 75. 15 = 3 × 5 and 25 = 5 × 5 share one 5, so the LCM is 3 × 5 × 5 = 75, not 15 × 25 = 375. Making one bus *less* frequent made the shared departures *rarer*, but not by as much as you might fear.",{"id":712,"type":172,"caption":713,"columns":714,"rows":718},"table-bus-what-if","Bus A every 15 minutes: how the meeting time depends on bus B (all computed)",[715,716,717],"Bus B every…","HCF(15, B)","LCM(15, B) = together every…",[719,722,725,728,731,733,736,737],[720,211,721],"10 min","30 min",[723,195,724],"12 min","60 min",[726,183,727],"14 min","210 min",[729,195,730],"18 min","90 min",[732,211,724],"20 min",[734,211,735],"25 min","75 min",[721,273,721],[738,273,738],"45 min",{"id":740,"type":80,"variant":361,"title":741,"markdown":742},"observation-bus","What the table shows","Bus B every 14 minutes shares no factor with 15, so they meet only every 210 minutes. Every 18 minutes is slower than every 14, yet they meet much more often (every 90 minutes), because 15 and 18 share a 3. Timetable planners choose gaps with big HCFs when they want services to connect often.",{"id":744,"type":143,"component":745,"componentVersion":5,"config":746,"objective":807,"textAlternative":808},"lab-sort-inv","sort-game",{"prompt":747,"bins":748,"items":758,"seconds":232},"Is each statement about whole numbers always true, sometimes true, or never true?",[749,752,755],{"id":750,"label":751},"always","Always true",{"id":753,"label":754},"sometimes","Sometimes true",{"id":756,"label":757},"never","Never true",[759,763,767,771,775,779,783,787,791,795,799,803],{"id":760,"label":761,"bin":750,"why":762},"i1","HCF(a, b) × LCM(a, b) = a × b","For two numbers the shared primes are counted twice on both sides.",{"id":764,"label":765,"bin":753,"why":766},"i2","LCM(a, b) = a × b","True only when HCF(a, b) = 1, e.g. 5 and 7; false for 4 and 6.",{"id":768,"label":769,"bin":750,"why":770},"i3","The HCF of two consecutive numbers is 1","A common factor would divide their difference, 1.",{"id":772,"label":773,"bin":756,"why":774},"i4","The HCF is bigger than both numbers","The HCF divides each number, so it cannot exceed either.",{"id":776,"label":777,"bin":750,"why":778},"i5","The HCF of two numbers divides their LCM","HCF divides a, and a divides the LCM.",{"id":780,"label":781,"bin":753,"why":782},"i6","HCF × LCM = product, for three numbers","Works for 3, 4, 5 (pairwise co-prime), fails for 2, 4, 8.",{"id":784,"label":785,"bin":756,"why":786},"i7","Two even numbers are co-prime","Both are divisible by 2, so their HCF is at least 2.",{"id":788,"label":789,"bin":753,"why":790},"i8","The HCF of two numbers equals one of them","Only when one divides the other, like 7 and 28.",{"id":792,"label":793,"bin":756,"why":794},"i9","The LCM is smaller than one of the numbers","Each number divides the LCM, so the LCM is at least as big as each.",{"id":796,"label":797,"bin":750,"why":798},"i10","Two consecutive even numbers have HCF 2","Both even, and the HCF must divide their difference 2.",{"id":800,"label":801,"bin":753,"why":802},"i11","Two composite numbers are co-prime","8 and 9 are; 4 and 6 are not.",{"id":804,"label":805,"bin":750,"why":806},"i12","Multiplying both numbers by 3 multiplies the HCF by 3","The extra 3 goes into both recipes, so into the shared part.","Sort statements about HCF and LCM into always, sometimes and never true, using examples and counterexamples.","This game has twelve statements to sort into \"Always true\", \"Sometimes true\" and \"Never true\".\n\nAlways true: HCF × LCM = a × b for two numbers; consecutive numbers have HCF 1; the HCF divides the LCM; two consecutive even numbers have HCF 2; multiplying both numbers by 3 multiplies the HCF by 3.\n\nSometimes true: LCM = a × b (only for co-primes, like 5 and 7, not 4 and 6); HCF × LCM = product for three numbers (true for 3, 4, 5 but not 2, 4, 8); the HCF equals one of the numbers (only when one divides the other); two composite numbers are co-prime (8 and 9 yes, 4 and 6 no).\n\nNever true: the HCF is bigger than both numbers; two even numbers are co-prime; the LCM is smaller than one of the numbers.\n\nFor \"sometimes\", give one example where it works and one where it fails.",{"id":810,"type":119,"title":811,"eyebrow":812,"navLabel":813},"ch10","Primes, squares and changing the story","Chapter 10","10 Primes and squares",{"id":815,"type":125,"prompt":816,"options":817,"explanation":825},"predict-prime-hcf","p is a prime number and n is any whole number. What can HCF(p, n) be?",[818,820,822,823],{"id":129,"label":819},"Any number up to p",{"id":132,"label":821},"Only 1 or p",{"id":135,"label":302},{"id":138,"label":824},"Always p","**Only 1 or p.** The HCF must divide p, and a prime has only two factors: 1 and itself. So HCF(p, n) = p if p divides n, and 1 otherwise. Test: HCF(7, 30) = 1, HCF(7, 42) = 7, HCF(13, 100) = 1, HCF(13, 65) = 13.",{"id":827,"type":172,"caption":828,"columns":829,"rows":832},"table-prime-hcf","HCF and LCM with a prime (all computed)",[830,831,179,177],"p, n","Does p divide n?",[833,836,838,841,845,848,851,853],[834,277,183,835],"7, 30","210",[837,264,265,417],"7, 42",[839,277,183,840],"11, 60","660",[842,264,843,844],"11, 121","11","121",[846,277,183,847],"13, 100","1300",[849,264,406,850],"13, 65","65",[852,264,211,211],"5, 5",[854,277,183,855],"2, 99","198",{"id":857,"type":125,"prompt":858,"options":859,"explanation":866},"predict-squares","Is the HCF of two **square numbers** (like 36 and 100) always a square number?",[860,861,863,864],{"id":129,"label":478},{"id":132,"label":862},"Only when both are even",{"id":135,"label":480},{"id":138,"label":865},"Only by luck","**Yes, always.** HCF(36, 100) = 4 = 2², HCF(16, 36) = 4 = 2², HCF(81, 144) = 9 = 3², HCF(100, 225) = 25 = 5². The reason: in a square number, every prime has an **even** power (36 = 2² × 3²). The HCF takes the smaller of two even powers, which is still even, so the HCF is a square too. The same argument shows the LCM of two squares is a square.",{"id":868,"type":172,"caption":869,"columns":870,"rows":874},"table-squares","Two squares: the HCF and LCM are squares too (all computed)",[871,179,177,872,873],"Squares","HCF is","LCM is",[875,879,883,888,891,896],[876,199,512,877,878],"36, 100","2²","30²",[880,199,881,877,882],"16, 36","144","12²",[884,375,885,886,887],"81, 144","1,296","3²","36²",[889,283,512,890,878],"100, 225","5²",[892,411,893,894,895],"64, 144","576","4²","24²",[897,183,898,899,900],"49, 25","1,225","1²","35²",{"id":902,"type":80,"variant":221,"title":903,"markdown":904},"aha-squares","A bonus pattern in the table","Look at the last two columns: HCF(a², b²) = HCF(a, b)² and LCM(a², b²) = LCM(a, b)². Squaring both numbers doubles every prime power, and min and max of doubled powers are just doubled min and max.",{"id":906,"type":80,"variant":907,"title":908,"markdown":909},"misconception-odd-coprime","misconception","\"Two odd numbers are always co-prime\"","Students who have just learned that consecutive odd numbers are co-prime often over-generalise: \"odd numbers don't share factors\". But HCF(15, 21) = 3, HCF(25, 35) = 5 and HCF(9, 27) = 9. Being odd only rules out the factor 2; other primes can still be shared. The investigation result was about **consecutive** odd numbers, whose difference is 2.",{"id":911,"type":43,"markdown":912},"changing-story","Investigating also means asking \"what if?\" about a word problem. A sweet shop packs **24 laddoos** with some barfis into identical boxes, using the most boxes possible. How does the number of boxes change with the number of barfis?\n\nThe number of boxes is HCF(24, barfis). With 36 barfis it is 12; with 30 barfis 6; with 25 barfis only 1, a single big box, because 24 and 25 are neighbours. With 48 barfis it is 24, because 24 divides 48. Tiny changes in a count can change the answer enormously.",{"id":914,"type":172,"caption":915,"columns":916,"rows":921},"table-laddoo-what-if","Boxes for 24 laddoos and different numbers of barfis (all computed)",[917,918,919,920],"Barfis","Most boxes = HCF(24, barfis)","Laddoos per box","Barfis per box",[922,923,924,926,927,928,929,931],[283,183,186,283],[210,218,199,211],[925,509,195,199],"32",[206,33,187,195],[278,509,195,211],[417,218,199,265],[930,186,183,187],"48",[496,33,187,211],{"id":933,"type":226,"itemId":934,"prompt":935,"check":936,"hints":938,"feedback":941},"practice-prime-lcm","hcf-and-lcm.investigate-prime-lcm","p = 17 and n = 51. What is LCM(p, n)?",{"kind":230,"answer":937,"tolerance":232},51,[939,940],"Does 17 divide 51?","If p divides n, the LCM is n.",{"correct":942,"incorrect":943},"Yes: 51 = 17 × 3, so HCF = 17 and LCM = 51.","17 divides 51, so the LCM is 51 itself.",{"id":945,"type":226,"itemId":946,"prompt":947,"check":948,"hints":949,"feedback":951},"practice-squares","hcf-and-lcm.investigate-hcf-squares","Without factorising the big numbers: HCF(12, 18) = 6. What is **HCF(144, 324)**? (144 = 12² and 324 = 18².)",{"kind":230,"answer":317,"tolerance":232},[950],"HCF(a², b²) = HCF(a, b)².",{"correct":952,"incorrect":953},"Yes: 6² = 36.","Squaring both numbers squares the HCF: 6² = 36.",{"id":955,"type":226,"itemId":956,"prompt":957,"check":958,"hints":961,"feedback":964},"practice-change-bus","hcf-and-lcm.investigate-bus-change","An auto stand sends autos to the station every **9 minutes** and to the market every **12 minutes**. If the market service changes to every **10 minutes**, how many minutes apart are the joint departures now?",{"kind":230,"answer":959,"tolerance":232,"unit":960},90,"minutes",[962,963],"Before the change they met every LCM(9, 12) = 36 minutes.","Find LCM(9, 10).",{"correct":965,"incorrect":966},"Yes: 9 and 10 are co-prime, so LCM = 90 minutes. The joint departures became much rarer (from every 36 to every 90 minutes).","LCM(9, 10) = 90: 9 and 10 share no factor, so the LCM is their product.",{"id":968,"type":119,"title":969,"eyebrow":970,"navLabel":971},"ch11","A grid investigation: the diagonal","Chapter 11","11 Diagonal puzzle",{"id":973,"type":43,"markdown":974},"diagonal-prose","Here is a classic investigation that seems to have nothing to do with HCF, until it does.\n\nDraw a rectangle on squared paper, **m** squares wide and **n** squares tall. Draw a straight line from the bottom-left corner to the top-right corner. **How many squares does the line pass through** (through the inside, not just touching a corner)?\n\nTry a 2 × 3 rectangle: the diagonal passes through 4 squares. A 4 × 6 rectangle: 8 squares, not twice as many. A 5 × 5 square: just 5, straight along the diagonal squares.",{"id":976,"type":125,"prompt":977,"options":978,"explanation":987},"predict-diagonal","The diagonal of a **2 × 3** rectangle crosses 4 squares. How many squares does the diagonal of a **4 × 6** rectangle cross?",[979,981,983,985],{"id":129,"label":980},"8, twice as many",{"id":132,"label":982},"10, which is 4 + 6",{"id":135,"label":984},"8, which is 4 + 6 − 2",{"id":138,"label":986},"24, every square","**8, which is 4 + 6 − 2.** A 4 × 6 rectangle is two 2 × 3 rectangles placed corner to corner along the diagonal, so the line crosses 2 × 4 = 8 squares. But look at the pattern in the table below: the answer is always **m + n − HCF(m, n)**. For 4 × 6: 4 + 6 − 2 = 8. Both explanations agree, which is a good sign.",{"id":989,"type":172,"caption":990,"columns":991,"rows":997},"table-diagonal","Squares crossed by the diagonal of an m × n rectangle (counted by computer)",[992,993,994,995,996],"m × n","m + n","HCF(m, n)","Squares crossed","m + n − HCF",[998,1000,1002,1004,1006,1008,1010,1012],[999,211,183,199,199],"2 × 3",[1001,265,183,218,218],"3 × 4",[1003,356,187,509,509],"4 × 6",[1005,356,211,211,211],"5 × 5",[1007,273,195,33,33],"6 × 9",[1009,607,187,33,33],"4 × 10",[1011,567,199,411,411],"8 × 12",[1013,210,218,186,186],"12 × 18",{"id":1015,"type":80,"variant":221,"title":1016,"markdown":1017},"aha-diagonal","Why the HCF appears","Moving along the diagonal, you enter a new square every time you cross a vertical grid line (m − 1 of them inside the rectangle) or a horizontal one (n − 1 of them). That would give 1 + (m − 1) + (n − 1) = m + n − 1 squares. But sometimes the line crosses a vertical and a horizontal line **at the same moment**, exactly at a grid corner, and that should only count once. The diagonal hits a grid corner inside the rectangle HCF(m, n) − 1 times (it passes through corners at 1⁄HCF, 2⁄HCF, … of the way along). Subtracting those gives m + n − 1 − (HCF − 1) = **m + n − HCF**.",{"id":1019,"type":226,"itemId":1020,"prompt":1021,"check":1022,"hints":1024,"feedback":1027},"practice-diagonal","hcf-and-lcm.investigate-diagonal-12-18","A rangoli grid is **12** squares by **18** squares. How many squares does a straight line from one corner to the opposite corner pass through?",{"kind":230,"answer":1023,"tolerance":232},24,[1025,1026],"Use m + n − HCF(m, n).","HCF(12, 18) = 6.",{"correct":1028,"incorrect":1029},"Yes: 12 + 18 − 6 = 24.","m + n − HCF = 12 + 18 − 6 = 24.",{"id":1031,"type":119,"title":1032,"eyebrow":1033,"navLabel":1034},"ch12","What did the investigation find?","Chapter 12","12 Findings",{"id":1036,"type":143,"component":1037,"componentVersion":5,"config":1038,"objective":1063,"textAlternative":1064},"lab-match-inv","match-pairs",{"prompt":1039,"mode":1040,"pairs":1041},"Match each investigation to what it found.","connect",[1042,1045,1047,1049,1052,1055,1058,1061],{"a":1043,"b":1044},"Product ÷ LCM of two numbers","Always equals the HCF",{"a":1046,"b":302},"HCF of consecutive numbers",{"a":1048,"b":304},"HCF of consecutive even numbers",{"a":1050,"b":1051},"HCF of two numbers 9 apart","Divides 9: 1, 3 or 9",{"a":1053,"b":1054},"HCF × LCM for 6, 10 and 15","30, but the product is 900",{"a":1056,"b":1057},"HCF(40, 60) compared with HCF(4, 6)","10 times bigger",{"a":1059,"b":1060},"Numbers ≤ 100 divisible by 4 and 6","8 (the multiples of 12)",{"a":1062,"b":635},"Smallest number > 2 with remainder 2 on ÷ 3, 4, 5","Match each investigation question to the pattern or answer you discovered.","This game connects eight investigation questions to their findings:\n\n- Product ÷ LCM of two numbers → always equals the HCF.\n- HCF of consecutive numbers → always 1.\n- HCF of consecutive even numbers → always 2.\n- HCF of two numbers 9 apart → divides 9, so it is 1, 3 or 9.\n- HCF × LCM for 6, 10, 15 → 1 × 30 = 30, but the product is 900.\n- HCF(40, 60) compared with HCF(4, 6) → 20 is 10 times 2.\n- Numbers up to 100 divisible by both 4 and 6 → 8, the multiples of LCM 12.\n- Smallest number above 2 leaving remainder 2 on division by 3, 4 and 5 → 62 = 60 + 2.",{"id":1066,"type":80,"variant":1067,"title":1068,"markdown":1069},"model-limit-inv","model_limit","Testing is not proving","Every pattern in this layer was found by testing examples, and most were then explained with a short reason. The explanations are what make them certain. The Deepen layer gives careful arguments for the product rule, for Euclid’s long division method and for the \"HCF divides the difference\" rule, so that you know they hold for **every** pair of whole numbers, not just the ones we tried.",{"id":1071,"type":1072,"title":1073,"questions":1074},"quiz-investigate","quiz","Investigation check",[1075,1088,1098,1108,1117,1129,1141,1150],{"itemId":1076,"prompt":1077,"options":1078,"correct":132,"why":1087},"hcf-and-lcm.investigate-q-counter","Which pair is a counterexample to \"LCM(a, b) is always a × b\"?",[1079,1081,1083,1085],{"id":129,"label":1080},"3 and 5",{"id":132,"label":1082},"6 and 8",{"id":135,"label":1084},"7 and 11",{"id":138,"label":1086},"1 and 9","LCM(6, 8) = 24, not 48. The others are co-prime, so their LCM is the product.",{"itemId":1089,"prompt":1090,"options":1091,"correct":129,"why":1097},"hcf-and-lcm.investigate-q-neighbours","HCF(1,000, 1,001) = ?",[1092,1093,1094,1095],{"id":129,"label":183},{"id":132,"label":265},{"id":135,"label":843},{"id":138,"label":1096},"1,000","Consecutive numbers always have HCF 1, because a common factor must divide their difference, 1.",{"itemId":1099,"prompt":1100,"options":1101,"correct":129,"why":1107},"hcf-and-lcm.investigate-q-diff","Two numbers differ by 4 and are both odd. Their HCF must be…",[1102,1103,1104,1105],{"id":129,"label":183},{"id":132,"label":187},{"id":135,"label":199},{"id":138,"label":1106},"1 or 2 or 4","The HCF divides 4, so it is 1, 2 or 4. Both numbers are odd, so 2 and 4 cannot divide them. It must be 1.",{"itemId":1109,"prompt":1110,"options":1111,"correct":135,"why":1116},"hcf-and-lcm.investigate-q-scale","HCF(6, 15) = 3. What is HCF(60, 150)?",[1112,1113,1114,1115],{"id":129,"label":195},{"id":132,"label":356},{"id":135,"label":210},{"id":138,"label":571},"Both numbers were multiplied by 10, so the HCF is multiplied by 10: 30.",{"itemId":1118,"prompt":1119,"options":1120,"correct":132,"why":1128},"hcf-and-lcm.investigate-q-three-works","For which triple does HCF × LCM equal the product?",[1121,1123,1125,1126],{"id":129,"label":1122},"2, 4, 6",{"id":132,"label":1124},"4, 5, 9",{"id":135,"label":511},{"id":138,"label":1127},"3, 6, 9","4, 5, 9 are pairwise co-prime: HCF = 1, LCM = 180 = 4 × 5 × 9.",{"itemId":1130,"prompt":1131,"options":1132,"correct":132,"why":1140},"hcf-and-lcm.investigate-q-remainder","Smallest number (other than 4) that leaves remainder 4 when divided by 6, 9 and 12?",[1133,1135,1136,1138],{"id":129,"label":1134},"22",{"id":132,"label":278},{"id":135,"label":1137},"112",{"id":138,"label":1139},"652","LCM(6, 9, 12) = 36, so 36 + 4 = 40.",{"itemId":1142,"prompt":1143,"options":1144,"correct":132,"why":1149},"hcf-and-lcm.investigate-q-count","How many numbers from 1 to 100 are divisible by both 6 and 9?",[1145,1146,1147,1148],{"id":129,"label":183},{"id":132,"label":211},{"id":135,"label":843},{"id":138,"label":411},"LCM(6, 9) = 18, and 100 ÷ 18 = 5 remainder 10, so 5: 18, 36, 54, 72, 90.",{"itemId":1151,"prompt":1152,"options":1153,"correct":138,"why":1161},"hcf-and-lcm.investigate-q-bus","Bus A runs every 12 min. Which gap for bus B makes them leave together **most** often?",[1154,1156,1158,1160],{"id":129,"label":1155},"11 min",{"id":132,"label":1157},"13 min",{"id":135,"label":1159},"16 min",{"id":138,"label":729},"LCM(12, 11) = 132, LCM(12, 13) = 156, LCM(12, 16) = 48, LCM(12, 18) = 36. The smallest LCM is 36, with 18 minutes.",{"id":1163,"type":1164,"title":1165,"points":1166},"cheat-sheet-investigate","summary","What the investigations found",[1167,1168,1169,1170,1171,1172,1173,1174,1175,1176],"**Product ÷ LCM = HCF** for two numbers, so HCF × LCM = a × b. LCM = a × b only when HCF = 1.","**If a divides b:** HCF = a and LCM = b.","**The HCF divides the difference** of the two numbers. Consecutive numbers: HCF 1. Consecutive odd: 1. Consecutive even: 2.","**The HCF divides the LCM**, and LCM ÷ HCF = (a ÷ HCF) × (b ÷ HCF).","**Three numbers:** HCF × LCM = product only when the numbers are pairwise co-prime. (6, 10, 15) is the classic counterexample.","**Scaling:** multiply both numbers by k and both the HCF and the LCM are multiplied by k.","**Same remainder r:** smallest number = LCM + r (bigger than the divisors); largest divisor = HCF of the numbers minus r.","**Same shortfall s:** when each remainder is s less than its divisor, the number is LCM − s.","**Counting:** numbers up to N divisible by both a and b = N ÷ LCM(a, b), rounded down.","**Diagonal of an m × n grid** passes through m + n − HCF(m, n) squares.",{"id":1178,"type":1179,"prompt":1180},"reflect-investigate","reflection","Make up your own conjecture about HCF or LCM (for example about square numbers, prime numbers, or numbers ending in 5). Test it on at least six examples, including one chosen to try to break it. Did it survive? Can you explain why?",{"id":1182,"type":1183,"conceptId":1184,"relation":1185,"explanation":1186},"conn-patterns-inv","connection","patterns","related_to","Common multiples repeat every LCM, and remainder answers form a pattern: 2, 62, 122, 182…",{"id":1188,"type":1183,"conceptId":1189,"relation":1185,"explanation":1190},"conn-properties-inv","properties-of-numbers","The rule that a common factor divides the difference is a divisibility property you can use far beyond HCF.",{"id":1192,"type":1183,"conceptId":1193,"relation":1194,"explanation":1195},"conn-prime-inv","prime-and-composite","helps_understand","Co-primes, twin primes and consecutive numbers all show up in the HCF investigations.",{"id":1197,"type":1198,"sourceIds":1199},"sources-investigate","sources",[1200,1201,1202,1203,1204,1205],"hcf-and-lcm-ncert-class6-playing-with-numbers","hcf-and-lcm-ncert-class10-real-numbers","hcf-and-lcm-wiki-gcd","hcf-and-lcm-wiki-lcm","hcf-and-lcm-khan-factors-multiples","hcf-and-lcm-nrich-factors-multiples-primes",[1200,1201,1202,1203,1204,1205],"needs_review",{"generatedBy":1209,"notes":1210},"claude-code","Draft generated with Python-checked arithmetic; pending owner review.","e99e7424e51eefd396b0ab8254809a5037e604ae357bf891f61489f5520b9cf4",{"component:hcf-lcm@1":1213,"logic:practice":1214,"component:sort-game@1":1215,"component:match-pairs@1":1216,"source:hcf-and-lcm-khan-factors-multiples":1217,"source:hcf-and-lcm-ncert-class10-real-numbers":1218,"source:hcf-and-lcm-ncert-class6-playing-with-numbers":1219,"source:hcf-and-lcm-nrich-factors-multiples-primes":1220,"source:hcf-and-lcm-wiki-gcd":1221,"source:hcf-and-lcm-wiki-lcm":1222},"83f068508b17b70184e45fbcf95d356790bfa2b0a98dcdb859f8e9dc105e5d33","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","01af9d960126bcdac368f15ef1b43edbf2e389c0ce966c7e903721b346655feb","48d1fd4c018916e9edeb2f76b15a8eb3cb83c72b08a9d3ab332a3c167fe9be0b","82c5fcc908046567f1a1bdb191f9b28fd42600389720d981bc1965d0c1542772","25674a9fbe152fe3c9ce7273cc44537dd3ebb9b393ed8ba01f12daf10e9d473f","857640343ebbf479db694438bf4af134fd5baee47f017b64110ded8b40bb2dc6","12412937f9bd209912735d662496fbe90c125f89d22844b5a753d934cc84ef2c",{"state":1224,"reviewer":1225,"selfReview":163,"reviewedAt":1226,"method":1227},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597413]