[{"data":1,"prerenderedAt":1099},["ShallowReactive",2],{"layer:hcf-and-lcm:discover":3},{"layer":4,"contentHash":1082,"dependencyHashes":1083,"approval":1093,"releaseId":1098},{"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":1077,"reviewStatus":1078,"authoring":1079},1,"hcf-and-lcm","en","discover","Sharing and meeting: meet the HCF and LCM","The biggest equal pieces and the next time things line up","Start from two puzzles, the biggest tile for a courtyard and the next time two lights flash together, and discover factors, multiples, common factors, common multiples, the HCF and the LCM, and how to tell which one a problem needs.",[13,14,15,16,17],"List the factors of a number using pairs, and the multiples of a number by skip-counting.","Find common factors and the HCF of two numbers by listing.","Find common multiples and the LCM of two numbers by listing.","Decide whether a real problem is about sharing (HCF) or meeting again (LCM).","Use the sense check HCF ≤ each number ≤ LCM to catch mistakes.",35,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Discover",{"label":26,"value":27},"Reading time","≈ 35 minutes",{"label":29,"value":30},"You need","Times tables up to 10 × 10",{"label":32,"value":33},"Chapters","11",{"label":35,"value":36},"Labs","Factor lists, frog race, sort game, memory match",{"label":38,"value":39},"Big ideas","HCF = sharing, LCM = meeting",[41,45,51,71,77,80,85,114,149,164,169,172,194,198,227,232,243,248,251,265,283,292,297,309,314,317,343,348,391,396,399,403,427,431,442,447,450,460,469,482,487,499,504,507,539,543,548,561,582,587,590,615,672,677,682,743,764,769,773,784,789,792,802,811,820,824,828,841,852,872,916,921,1035,1049,1053,1059,1064,1068],{"id":42,"type":43,"markdown":44},"intro-two-puzzles","prose","Here are two puzzles that look completely different.\n\n**Puzzle 1.** Meera's grandmother wants to tile a courtyard that is **240 cm long and 180 cm wide** with square tiles, all the same size, with no cutting and no gaps. She wants the tiles to be as **big** as possible, so there are fewer joins to clean. How big should each tile be?\n\n**Puzzle 2.** At Diwali, two strings of lights are switched on at the same moment. One string flashes every **4 seconds**, the other every **6 seconds**. After how many seconds do they flash **together** again?\n\nThe first puzzle is about **cutting something into equal pieces**. The second is about **two repeating things meeting again**. By the end of this layer you will be able to solve both in your head, and you will know the two special numbers hiding inside them: the **HCF** and the **LCM**.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"big-question","callout","question","The question for this topic","When will two blinking lights flash together again, and what is the biggest tile that fits a floor exactly?\n\nKeep both puzzles in mind. We will come back and solve them properly in Chapters 3 and 6.",{"id":52,"type":53,"tone":54,"items":55},"spec-two-ideas","spec","blue",[56,60,64,68],{"label":57,"big":58,"value":59},"HCF","Share \u002F cut","Highest Common Factor: the **biggest** number that divides two numbers exactly. Used when you split things into the largest equal parts.",{"label":61,"big":62,"value":63},"LCM","Meet \u002F match","Lowest Common Multiple: the **smallest** number that two numbers both divide into. Used when repeating things line up again.",{"label":65,"big":66,"value":67},"Built from","Factors","HCF is built from **factors**: the numbers that go into a number.",{"label":65,"big":69,"value":70},"Multiples","LCM is built from **multiples**: the numbers a number goes into (its times table).",{"id":72,"type":73,"title":74,"eyebrow":75,"navLabel":76},"ch01","chapter","Factors: the numbers that fit exactly","Chapter 01","1 Factors",{"id":78,"type":43,"markdown":79},"factors-meaning","Take 12 marbles. Can you put them into equal rows with none left over?\n\n- 1 row of 12 ✓\n- 2 rows of 6 ✓\n- 3 rows of 4 ✓\n- 4 rows of 3 ✓\n- 6 rows of 2 ✓\n- 12 rows of 1 ✓\n- 5 rows? 12 ÷ 5 = 2 remainder 2. ✗ Two marbles are left over.\n\nThe numbers that worked, **1, 2, 3, 4, 6, 12**, are the **factors** of 12. A factor of a number divides it **exactly**, with **no remainder**.\n\nNotice they came in pairs: 1 and 12, 2 and 6, 3 and 4. Each pair multiplies to 12. That pairing trick is the fastest way to list factors without missing any.",{"id":81,"type":47,"variant":82,"title":83,"markdown":84},"def-factor","definition","Factor","A **factor** of a whole number is a whole number that divides it exactly, leaving remainder 0.\n\n5 is a factor of 20, because 20 ÷ 5 = 4 with nothing left over. 6 is **not** a factor of 20, because 20 ÷ 6 = 3 remainder 2.\n\nEvery number has **1** and **itself** as factors.",{"id":86,"type":87,"title":88,"items":89},"steps-pairs","steps","Listing factors without missing any: the pairs method",[90,94,98,102,106,110],{"title":91,"tag":92,"text":93},"Start with 1","1 × 18","Every number is 1 × itself. Write 1 on the left and 18 on the right.",{"title":95,"tag":96,"text":97},"Try 2","2 × 9","18 ÷ 2 = 9 exactly, so 2 and 9 are both factors.",{"title":99,"tag":100,"text":101},"Try 3","3 × 6","18 ÷ 3 = 6 exactly, so 3 and 6 are both factors.",{"title":103,"tag":104,"text":105},"Try 4 and 5","no","18 ÷ 4 and 18 ÷ 5 leave remainders, so neither is a factor.",{"title":107,"tag":108,"text":109},"Stop when you meet","6 already found","The next number to try is 6, which is already on the right. The pairs have met in the middle, so you are done.",{"title":111,"tag":112,"text":113},"Read the list","6 factors","Factors of 18: **1, 2, 3, 6, 9, 18**.",{"id":115,"type":116,"caption":117,"columns":118,"rows":121},"table-factors","table","Factors of some everyday numbers (each list found with the pairs method)",[119,66,120],"Number","How many",[122,126,130,133,136,139,142,145],[123,124,125],"8","1, 2, 4, 8","4",[127,128,129],"12","1, 2, 3, 4, 6, 12","6",[131,132,125],"15","1, 3, 5, 15",[134,135,129],"18","1, 2, 3, 6, 9, 18",[137,138,129],"20","1, 2, 4, 5, 10, 20",[140,141,123],"24","1, 2, 3, 4, 6, 8, 12, 24",[143,144,123],"30","1, 2, 3, 5, 6, 10, 15, 30",[146,147,148],"36","1, 2, 3, 4, 6, 9, 12, 18, 36","9",{"id":150,"type":151,"itemId":152,"prompt":153,"check":154,"hints":158,"feedback":161},"practice-factors-24","practice","hcf-and-lcm.discover-count-factors-24","How many factors does **24** have? (Count 1 and 24 too.)",{"kind":155,"answer":156,"tolerance":157},"number",8,0,[159,160],"Use pairs: 1 × 24, 2 × 12, …","Keep trying 3, 4, 5 until the pairs meet.",{"correct":162,"incorrect":163},"Yes: 1, 2, 3, 4, 6, 8, 12, 24. That is 8 factors in 4 pairs.","List the pairs: 1 × 24, 2 × 12, 3 × 8, 4 × 6. That gives 8 factors: 1, 2, 3, 4, 6, 8, 12, 24.",{"id":165,"type":73,"title":166,"eyebrow":167,"navLabel":168},"ch02","Common factors and the Highest Common Factor","Chapter 02","2 Common factors",{"id":170,"type":43,"markdown":171},"common-factors-prose","Now take **two** numbers, 12 and 18, and write their factors side by side.\n\n- Factors of 12: 1, 2, 3, 4, 6, 12\n- Factors of 18: 1, 2, 3, 6, 9, 18\n\nSome numbers appear in **both** lists: **1, 2, 3, 6**. These are the **common factors** of 12 and 18. \"Common\" here means *shared*, just as a common room is a room everyone shares.\n\nThe **highest** (biggest) of the common factors is **6**. It is called the **Highest Common Factor**, or **HCF**, of 12 and 18.\n\nWe write this as **HCF(12, 18) = 6**.",{"id":173,"type":116,"caption":174,"columns":175,"rows":179},"table-common-12-18","Circle the shared factors: 12 and 18",[83,176,177,178],"Divides 12?","Divides 18?","Common?",[180,184,186,188,190,191,192,193],[181,182,182,183],"1","yes","**yes**",[185,182,182,183],"2",[187,182,182,183],"3",[125,182,104,189],"—",[129,182,182,183],[148,104,182,189],[127,182,104,189],[134,104,182,189],{"id":195,"type":47,"variant":82,"title":196,"markdown":197},"names-hcf","Three names, one idea","The **Highest Common Factor (HCF)** is the biggest number that is a factor of every number in the group.\n\nYou will also see it called:\n- **GCD**, the **Greatest Common Divisor** (a divisor is just another word for a factor), and\n- **GCF**, the **Greatest Common Factor**.\n\nAll three names mean exactly the same thing. Indian textbooks usually say HCF.",{"id":199,"type":200,"component":201,"componentVersion":5,"config":202,"objective":220,"textAlternative":221,"help":222},"lab-lists","interactive","hcf-lcm",{"sets":203,"views":217,"challenge":219},[204,207,208,211,214],[205,206],12,18,[156,205],[209,210],16,24,[212,213],15,25,[215,216],9,14,[218],"lists",false,"List the factors of two numbers, spot the ones they share, and pick out the highest common factor.","This lab shows two factor lists side by side and lights up the factors that appear in both.\n\n- **12 and 18:** factors of 12 are 1, 2, 3, 4, 6, 12; of 18 are 1, 2, 3, 6, 9, 18. Shared: 1, 2, 3, 6. HCF = **6**.\n- **8 and 12:** factors of 8 are 1, 2, 4, 8; of 12 are 1, 2, 3, 4, 6, 12. Shared: 1, 2, 4. HCF = **4**.\n- **16 and 24:** shared factors 1, 2, 4, 8. HCF = **8**.\n- **15 and 25:** shared factors 1, 5. HCF = **5**.\n- **9 and 14:** factors of 9 are 1, 3, 9; of 14 are 1, 2, 7, 14. The only shared factor is 1, so HCF = **1**.\n\nWhat to notice: 1 is always a common factor. Every other common factor is also a factor of the HCF. For 12 and 18, the common factors 1, 2, 3 and 6 are exactly the factors of 6.",{"simplerExplanation":223,"hints":224},"Write both lists. Put a ring round every number that is in both. The biggest ringed number is the HCF.",[225,226],"1 is always in both lists.","The HCF can never be bigger than the smaller number.",{"id":228,"type":47,"variant":229,"title":230,"markdown":231},"aha-common-factors","aha","The common factors are the factors of the HCF","Look again at 12 and 18. Their common factors are 1, 2, 3, 6, and those are exactly the factors of 6, the HCF.\n\nThis always happens. Once you know the HCF, you know **every** common factor for free: they are just the factors of the HCF. You will see why in the Deepen layer.",{"id":233,"type":151,"itemId":234,"prompt":235,"check":236,"hints":237,"feedback":240},"practice-hcf-16-24","hcf-and-lcm.discover-hcf-16-24","What is the HCF of **16** and **24**?",{"kind":155,"answer":156,"tolerance":157},[238,239],"List the factors of 16 first.","Which is the biggest one that also divides 24?",{"correct":241,"incorrect":242},"Right. Common factors are 1, 2, 4, 8, and the highest is 8.","Factors of 16: 1, 2, 4, 8, 16. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Shared: 1, 2, 4, 8. The HCF is 8.",{"id":244,"type":73,"title":245,"eyebrow":246,"navLabel":247},"ch03","Using the HCF: sharing and cutting","Chapter 03","3 Using the HCF",{"id":249,"type":43,"markdown":250},"hcf-use-prose","The HCF appears whenever you want to **split things into equal groups or equal pieces, as big as possible, with nothing left over**.\n\nThe key words are *largest*, *greatest*, *maximum*, *longest*, *as many as possible (identical groups)*, and a feeling of **cutting, dividing, sharing or packing**. The answer is always **smaller than or equal to** the numbers you started with, because you are breaking them down.",{"id":252,"type":253,"title":254,"problem":255,"steps":256,"help":263},"we-sweets","worked_example","Packing sweets for a family function","A sweet shop in Jaipur has **24 laddoos** and **36 barfis**. The owner wants to pack them into identical boxes, each box with the same number of laddoos and the same number of barfis, with nothing left over. What is the **greatest number of boxes** she can make, and what goes in each box?",[257,258,259,260,261,262],"Each box must hold the same share, so the number of boxes must divide **both** 24 and 36 exactly.","Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.","Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36.","Common factors: 1, 2, 3, 4, 6, 12. The greatest is **12**.","So she can make **12 boxes**. Each has 24 ÷ 12 = **2 laddoos** and 36 ÷ 12 = **3 barfis**.","Check: 12 × 2 = 24 ✓ and 12 × 3 = 36 ✓.",{"simplerExplanation":264},"The number of boxes has to share out both kinds of sweet exactly, so it must be a common factor. The biggest common factor gives the most boxes.",{"id":266,"type":267,"prompt":268,"options":269,"explanation":282},"predict-tile","prediction","Back to Puzzle 1. The courtyard is **240 cm × 180 cm**. Which is the **largest** square tile that fits exactly, with no cutting?",[270,273,276,279],{"id":271,"label":272},"a","20 cm",{"id":274,"label":275},"b","30 cm",{"id":277,"label":278},"c","60 cm",{"id":280,"label":281},"d","90 cm","**60 cm.** The side of the tile must divide 240 exactly (to fit along the length) **and** 180 exactly (to fit along the width). So it must be a common factor, and the largest one is HCF(240, 180) = 60.\n\n20 cm and 30 cm also fit, but they are smaller. 90 cm does not fit along 240 cm (240 ÷ 90 leaves 60). With 60 cm tiles you need 4 tiles along and 3 across: **12 tiles** in all.",{"id":284,"type":253,"title":285,"problem":286,"steps":287},"we-ribbon","Cutting ribbons for rakhis","Arjun has two ribbons, **18 m** and **24 m** long. He wants to cut both into pieces that are all the **same length**, as **long as possible**, with no ribbon wasted. How long is each piece, and how many pieces does he get?",[288,289,290,291],"The piece length must divide 18 and 24 exactly, so it is a common factor.","Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.","Common factors: 1, 2, 3, 6. The longest piece is **6 m**.","Pieces: 18 ÷ 6 = 3 and 24 ÷ 6 = 4, so **7 pieces** in all.",{"id":293,"type":47,"variant":294,"title":295,"markdown":296},"try-it-paper","try_it","Tile a rectangle with paper squares","Draw a rectangle 12 squares long and 8 squares wide on squared paper.\n\n1. Try to cover it with 2 × 2 squares. Does it work?\n2. Try 3 × 3 squares. What goes wrong?\n3. Try 4 × 4 squares. Does it work? Try 8 × 8.\n\nThe biggest square that works is 4 × 4, and 4 is the HCF of 12 and 8. Every square that works (1, 2 and 4) is a common factor.",{"id":298,"type":151,"itemId":299,"prompt":300,"check":301,"hints":303,"feedback":306},"practice-rows","hcf-and-lcm.discover-rows-students","A school has **32 boys** and **40 girls** for a drill display. They must stand in rows, every row the same size, and each row must be all boys or all girls. What is the **largest** possible number of students in each row?",{"kind":155,"answer":156,"tolerance":157,"unit":302},"students",[304,305],"The row size must divide 32 and also divide 40.","Find the HCF of 32 and 40.",{"correct":307,"incorrect":308},"Yes: HCF(32, 40) = 8. That gives 4 rows of boys and 5 rows of girls.","The row size must be a common factor of 32 and 40. Common factors: 1, 2, 4, 8. The largest is 8.",{"id":310,"type":73,"title":311,"eyebrow":312,"navLabel":313},"ch04","Multiples: the skip-counting trail","Chapter 04","4 Multiples",{"id":315,"type":43,"markdown":316},"multiples-prose","Now for the other idea. Count in 4s: **4, 8, 12, 16, 20, 24, 28, 32, 36, 40**, … These are the **multiples** of 4. They are just the 4 times table, and they go on for ever.\n\nA **multiple** of a number is what you get when you multiply it by 1, 2, 3, 4, and so on. So 20 is a multiple of 4 (because 4 × 5 = 20), but 22 is not.\n\nFactors and multiples are two sides of one fact. **4 × 5 = 20** tells you that 4 and 5 are *factors* of 20, and that 20 is a *multiple* of 4 and of 5.",{"id":318,"type":116,"caption":319,"columns":320,"rows":324},"table-factor-vs-multiple","Factors and multiples: the same fact, seen from two ends",[321,322,323],"Compare","Factors of a number","Multiples of a number",[325,329,332,336,340],[326,327,328],"Direction","Go **into** the number","The number goes **into** them",[330,128,331],"Example for 12","12, 24, 36, 48, 60, …",[333,334,335],"How many?","A limited, countable list","Never ends",[337,338,339],"Smallest","Always 1","The number itself",[341,339,342],"Biggest","None: they go on for ever",{"id":344,"type":47,"variant":345,"title":346,"markdown":347},"misconception-fm","misconception","\"Factors and multiples are the same thing\"","They are opposite directions of the same multiplication. A **factor** is smaller than or equal to the number; a **multiple** is bigger than or equal to it.\n\nMemory trick: **F**actors **F**it inside; **M**ultiples **M**ake bigger numbers. And every number is both a factor and a multiple of itself: 12 is a factor of 12 and a multiple of 12.",{"id":349,"type":350,"title":351,"terms":352},"glossary-discover-1","glossary","Words for this chapter and the last",[353,357,361,365,369,373,377,380,384,387],{"term":354,"meaning":355,"example":356},"factor","A whole number that divides another whole number exactly, with no remainder.","The factors of 10 are 1, 2, 5 and 10.",{"term":358,"meaning":359,"example":360},"divisor","Another word for a factor: a number that divides another exactly.","3 is a divisor of 21.",{"term":362,"meaning":363,"example":364},"divisible","A number is divisible by another if it can be divided by it with no remainder.","36 is divisible by 9.",{"term":366,"meaning":367,"example":368},"multiple","A number you get by multiplying a given number by 1, 2, 3, …","Multiples of 7: 7, 14, 21, 28, …",{"term":370,"meaning":371,"example":372},"common factor","A number that is a factor of two or more numbers at once.","3 is a common factor of 12 and 18.",{"term":374,"meaning":375,"example":376},"common multiple","A number that is a multiple of two or more numbers at once.","24 is a common multiple of 6 and 8.",{"term":57,"meaning":378,"example":379},"Highest Common Factor: the greatest number that divides all the given numbers exactly.","HCF(12, 18) = 6.",{"term":381,"meaning":382,"example":383},"GCD \u002F GCF","Greatest Common Divisor \u002F Greatest Common Factor: other names for the HCF.","GCD(8, 12) = 4.",{"term":61,"meaning":385,"example":386},"Lowest (or Least) Common Multiple: the smallest number that is a multiple of all the given numbers.","LCM(4, 6) = 12.",{"term":388,"meaning":389,"example":390},"remainder","What is left over after dividing as many times as possible.","17 ÷ 5 = 3 remainder 2.",{"id":392,"type":73,"title":393,"eyebrow":394,"navLabel":395},"ch05","Common multiples and the Lowest Common Multiple","Chapter 05","5 Common multiples",{"id":397,"type":43,"markdown":398},"common-multiples-prose","Write the multiples of 4 and of 6 side by side:\n\n- Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, …\n- Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, …\n\nSome numbers are on **both** trails: **12, 24, 36**, … These are the **common multiples** of 4 and 6. Unlike common factors, they never run out: 12, 24, 36, 48, 60 and so on.\n\nThe **lowest** (smallest) common multiple is **12**. It is called the **Lowest Common Multiple**, or **LCM**. We write **LCM(4, 6) = 12**.\n\nLook at the list of common multiples again: 12, 24, 36, 48… They are all multiples of 12. Once you know the LCM, every other common multiple is just a multiple of it.",{"id":400,"type":47,"variant":82,"title":401,"markdown":402},"def-lcm","Lowest Common Multiple (LCM)","The **LCM** of two or more numbers is the **smallest** number (other than 0) that is a multiple of every one of them.\n\nSome books say **Least** Common Multiple. Lowest and least mean the same here.\n\nWe ignore 0 on purpose: 0 = 4 × 0 = 6 × 0, so technically 0 is a multiple of everything, which would make it a useless answer.",{"id":404,"type":200,"component":201,"componentVersion":5,"config":405,"objective":420,"textAlternative":421,"help":422},"lab-frogs",{"sets":406,"views":417,"challenge":419},[407,410,412,414,416],[408,409],2,3,[409,411],4,[411,413],6,[409,415],5,[413,156],[418],"frogs",true,"Watch two frogs hop along a line of stones at different step sizes and predict the first stone they both land on.","In this game two frogs start together at stone 0 on a long line of numbered stones. Each frog always takes jumps of the same size. The stones each frog lands on are the multiples of its jump size. The first stone both frogs land on is the LCM.\n\n- **Jumps of 2 and 3:** frog A lands on 2, 4, **6**, 8…; frog B on 3, **6**, 9… First shared stone: **6**.\n- **Jumps of 3 and 4:** 3, 6, 9, **12** and 4, 8, **12**. First shared stone: **12**.\n- **Jumps of 4 and 6:** 4, 8, **12** and 6, **12**. First shared stone: **12**, not 24.\n- **Jumps of 3 and 5:** first shared stone **15**.\n- **Jumps of 6 and 8:** 6, 12, 18, **24** and 8, 16, **24**. First shared stone: **24**, not 48.\n\nIn challenge mode you predict the meeting stone before the frogs hop. After the first meeting, the frogs keep meeting at every multiple of that stone: for jumps 4 and 6, at 12, 24, 36…",{"simplerExplanation":423,"hints":424},"Each frog lands on its own times table. The first number that is in both times tables is where they meet: the LCM.",[425,426],"Count along the bigger jump: 6, 12, 18… and stop at the first one the smaller jump also reaches.","Multiplying the two jumps always gives a meeting stone, but not always the first one.",{"id":428,"type":47,"variant":345,"title":429,"markdown":430},"misconception-multiply","\"The LCM is just the two numbers multiplied\"","Sometimes it is: LCM(3, 5) = 15 = 3 × 5. But often it is not: 4 × 6 = 24, yet LCM(4, 6) = **12**. The product is always *a* common multiple, but it is only the **lowest** one when the two numbers share no factor except 1.\n\nIn the frogs lab, jumps of 4 and 6 meet at 12, long before 24.",{"id":432,"type":151,"itemId":433,"prompt":434,"check":435,"hints":436,"feedback":439},"practice-lcm-6-8","hcf-and-lcm.discover-lcm-6-8","What is the LCM of **6** and **8**?",{"kind":155,"answer":210,"tolerance":157},[437,438],"List multiples of 8: 8, 16, 24, …","Which is the first one that 6 also divides?",{"correct":440,"incorrect":441},"Yes: 24 is the first number in both times tables. LCM(6, 8) = 24.","Multiples of 8: 8, 16, 24. Is 8 a multiple of 6? No. 16? No. 24 = 6 × 4? Yes. So LCM(6, 8) = 24. (6 × 8 = 48 is a common multiple, but not the lowest.)",{"id":443,"type":73,"title":444,"eyebrow":445,"navLabel":446},"ch06","Using the LCM: when things meet again","Chapter 06","6 Using the LCM",{"id":448,"type":43,"markdown":449},"lcm-use-prose","The LCM appears whenever **two or more repeating things start together and you want to know when they will next line up**: lights flashing, bells ringing, buses leaving, runners lapping, or just the smallest number that several numbers all divide.\n\nThe key words are *together again*, *at the same time*, *next*, *first time*, *smallest* or *least (number that…)*, and a feeling of **repeating, cycles and meeting**. The answer is always **bigger than or equal to** the numbers you started with, because you are waiting for both cycles to finish.",{"id":451,"type":253,"title":452,"problem":453,"steps":454},"we-lights","Solving Puzzle 2: the Diwali lights","One string of lights flashes every **4 seconds**, another every **6 seconds**. They flash together at the start. When do they next flash together?",[455,456,457,458,459],"The first string flashes at 4, 8, 12, 16, 20, 24 seconds, and so on: the multiples of 4.","The second flashes at 6, 12, 18, 24 seconds: the multiples of 6.","The first time in both lists is **12 seconds**, so that is LCM(4, 6).","They will then flash together every 12 seconds: at 12, 24, 36, 48 … seconds.","In one minute (60 seconds) they flash together 5 times after the start, at 12, 24, 36, 48 and 60 seconds.",{"id":461,"type":253,"title":462,"problem":463,"steps":464},"we-buses","Two buses at the depot","At a bus stand in Chennai, the airport bus leaves every **15 minutes** and the beach bus every **20 minutes** (an imagined timetable, not a real one). Both leave together at **8:00 a.m.** When do they next leave together?",[465,466,467,468],"Airport bus: 8:00, then every 15 minutes. Minutes after 8:00: 15, 30, 45, 60.","Beach bus minutes after 8:00: 20, 40, 60.","The first shared time is **60 minutes** after 8:00, which is LCM(15, 20).","So they next leave together at **9:00 a.m.**, and then every hour after that.",{"id":470,"type":267,"prompt":471,"options":472,"explanation":481},"predict-bells","Two school bells are tested together at 9:00. One rings every **6 minutes**, the other every **9 minutes**. When do they next ring together?",[473,475,477,479],{"id":271,"label":474},"9:03, because 9 − 6 = 3",{"id":274,"label":476},"9:15, because 6 + 9 = 15",{"id":277,"label":478},"9:18",{"id":280,"label":480},"9:54, because 6 × 9 = 54","**9:18.** Multiples of 6: 6, 12, 18… Multiples of 9: 9, 18… The first shared one is 18, so LCM(6, 9) = 18 minutes. Subtracting or adding the numbers has nothing to do with it. Multiplying gives 54, which *is* a time they ring together (18 × 3 = 54), but not the first.",{"id":483,"type":47,"variant":484,"title":485,"markdown":486},"example-cricket","example","Nets practice schedule","A cricket academy runs batting drills every **3 days** and fielding drills every **4 days**. Both happened on 1 June. They happen on the same day again after LCM(3, 4) = **12 days**, on 13 June, then on 25 June.",{"id":488,"type":151,"itemId":489,"prompt":490,"check":491,"hints":493,"feedback":496},"practice-runners","hcf-and-lcm.discover-runners","Two friends jog round a park. Kavya takes **8 minutes** per round and Rohan takes **12 minutes**. They start together at the gate. After how many minutes are they next at the gate together?",{"kind":155,"answer":210,"tolerance":157,"unit":492},"minutes",[494,495],"Kavya is at the gate at 8, 16, 24, … minutes.","When is Rohan at the gate?",{"correct":497,"incorrect":498},"Yes: LCM(8, 12) = 24 minutes. Kavya has done 3 rounds and Rohan 2.","Kavya: 8, 16, 24… Rohan: 12, 24… First shared time: 24 minutes.",{"id":500,"type":73,"title":501,"eyebrow":502,"navLabel":503},"ch07","HCF is small, LCM is big","Chapter 07","7 Small and big",{"id":505,"type":43,"markdown":506},"small-big-prose","Put the two answers for 12 and 18 next to each other:\n\n- HCF(12, 18) = **6**. It is **at most** the smaller number, 12.\n- LCM(12, 18) = **36**. It is **at least** the bigger number, 18.\n\nThis gives you a quick sense check. If you are asked for an HCF and your answer is bigger than one of the numbers, something went wrong. If you are asked for an LCM and your answer is smaller than one of the numbers, something went wrong.\n\nThe HCF **divides** each number. Each number **divides** the LCM. So the HCF always sits at the bottom and the LCM at the top, with the numbers themselves in between.",{"id":508,"type":116,"caption":509,"columns":510,"rows":513},"table-small-big","HCF and LCM side by side (every value checked by computer)",[511,57,61,512],"Numbers","Sense check",[514,517,520,523,527,530,534],[515,185,127,516],"4 and 6","2 ≤ 4 and 12 ≥ 6 ✓",[518,129,146,519],"12 and 18","6 ≤ 12 and 36 ≥ 18 ✓",[521,125,140,522],"8 and 12","4 ≤ 8 and 24 ≥ 12 ✓",[524,181,525,526],"5 and 7","35","1 ≤ 5 and 35 ≥ 7 ✓",[528,129,127,529],"6 and 12","6 ≤ 6 and 12 ≥ 12 ✓",[531,187,532,533],"9 and 15","45","3 ≤ 9 and 45 ≥ 15 ✓",[535,536,537,538],"10 and 25","5","50","5 ≤ 10 and 50 ≥ 25 ✓",{"id":540,"type":47,"variant":345,"title":541,"markdown":542},"misconception-bigger","\"Highest means the HCF must be the bigger answer\"","The word *highest* in HCF tricks many learners. It is the highest of the **common factors**, which are all small. The word *lowest* in LCM is the lowest of the **common multiples**, which are all big. So the \"highest\" one is the smaller answer and the \"lowest\" one is the bigger answer!",{"id":544,"type":47,"variant":545,"title":546,"markdown":547},"observation-special","observation","Two special cases to spot","- **One number divides the other.** For 6 and 12: HCF = **6** (the smaller) and LCM = **12** (the bigger). Check: HCF(6, 12) = 6, LCM(6, 12) = 12.\n- **The numbers share no factor except 1.** For 5 and 7: HCF = **1** and LCM = **35**, the product. Check: HCF(5, 7) = 1, LCM(5, 7) = 35. Such numbers are called **co-primes**.",{"id":549,"type":267,"prompt":550,"options":551,"explanation":560},"predict-co-prime","Without listing anything, what are the HCF and LCM of **8** and **9**?",[552,554,556,558],{"id":271,"label":553},"HCF 1, LCM 72",{"id":274,"label":555},"HCF 1, LCM 17",{"id":277,"label":557},"HCF 8, LCM 9",{"id":280,"label":559},"HCF 3, LCM 24","**HCF 1, LCM 72.** 8 = 2 × 2 × 2 and 9 = 3 × 3 share no factor except 1, so they are co-prime. For co-primes the HCF is 1 and the LCM is the product: 8 × 9 = 72. Checked: HCF(8, 9) = 1 and LCM(8, 9) = 72. Neither 8 nor 9 is prime, yet they are co-prime.",{"id":562,"type":151,"itemId":563,"prompt":564,"check":565,"hints":577,"feedback":579},"practice-sense-check","hcf-and-lcm.discover-sense-check","Priya says the HCF of 20 and 30 is 60. Without calculating, how can you tell she is wrong?",{"kind":566,"options":567,"correct":576},"choice",[568,570,572,574],{"id":271,"label":569},"The HCF cannot be bigger than 20, the smaller number",{"id":274,"label":571},"60 is an odd number",{"id":277,"label":573},"The HCF is always 1",{"id":280,"label":575},"She is right",[271],[578],"A factor of 20 cannot be bigger than 20.",{"correct":580,"incorrect":581},"Exactly. 60 is the LCM, not the HCF. HCF(20, 30) = 10.","The HCF must divide 20, so it cannot be more than 20. Priya found the LCM (60) by mistake. HCF(20, 30) = 10.",{"id":583,"type":73,"title":584,"eyebrow":585,"navLabel":586},"ch08","Sharing or meeting? Choosing HCF or LCM","Chapter 08","8 HCF or LCM?",{"id":588,"type":43,"markdown":589},"choose-prose","Most word problems about HCF and LCM are easy once you decide **which one** you need. Ask yourself one question:\n\n> **Am I breaking things into equal parts, or waiting for repeating things to line up?**\n\nIf you are breaking, sharing, cutting, packing or arranging into the **biggest** equal groups, you want the **HCF**. The answer will be *smaller* than the numbers.\n\nIf you are waiting for things to happen **together**, or looking for the **smallest** number that several numbers all divide, you want the **LCM**. The answer will be *bigger* than the numbers.",{"id":591,"type":116,"caption":592,"columns":593,"rows":596},"table-clues","Clue words and the idea behind them",[594,595,29],"Clue in the problem","Picture it as",[597,600,603,606,609,612],[598,599,57],"largest tile, longest piece, biggest box","Cutting into equal pieces",[601,602,57],"greatest number of identical groups","Sharing everything out",[604,605,57],"maximum students in each row","Arranging in equal rows",[607,608,61],"ring together, flash together, meet again","Cycles lining up",[610,611,61],"smallest number divisible by …","First shared stop on the trails",[613,614,61],"least number of sweets that can be shared equally among 4, 6 or 8 children","A number every group size divides",{"id":616,"type":200,"component":617,"componentVersion":5,"config":618,"objective":668,"textAlternative":669,"help":670},"lab-sort-basic","sort-game",{"prompt":619,"bins":620,"items":627,"seconds":157},"Is each problem about sharing into the biggest equal parts (HCF) or about things meeting again (LCM)?",[621,624],{"id":622,"label":623},"hcf","HCF problem",{"id":625,"label":626},"lcm","LCM problem",[628,632,636,640,644,648,652,656,660,664],{"id":629,"label":630,"bin":622,"why":631},"s1","Largest square tile to cover a 240 cm × 180 cm floor","The tile side must divide both lengths exactly, and you want the biggest: HCF.",{"id":633,"label":634,"bin":625,"why":635},"s2","Two lights flash every 4 s and 6 s. When do they flash together?","Two repeating flashes lining up again: the first shared multiple, LCM = 12 s.",{"id":637,"label":638,"bin":622,"why":639},"s3","Pack 24 laddoos and 36 barfis into the most identical boxes","The number of boxes must divide both amounts. Most boxes = HCF = 12.",{"id":641,"label":642,"bin":625,"why":643},"s4","Buses leave every 15 and 20 minutes. When do they leave together?","Two timetables meeting: LCM(15, 20) = 60 minutes.",{"id":645,"label":646,"bin":622,"why":647},"s5","Longest equal pieces from ribbons of 18 m and 24 m","Cutting both into the longest equal pieces: HCF = 6 m.",{"id":649,"label":650,"bin":625,"why":651},"s6","Smallest number that both 6 and 8 divide exactly","A number both divide is a common multiple; smallest means LCM = 24.",{"id":653,"label":654,"bin":622,"why":655},"s7","Most students per row with 32 boys and 40 girls in single-gender equal rows","Row size must divide both 32 and 40: HCF = 8.",{"id":657,"label":658,"bin":625,"why":659},"s8","Two friends jog laps of 8 and 12 minutes. When are they at the gate together?","Laps repeat; they meet at the first shared multiple, 24 minutes.",{"id":661,"label":662,"bin":622,"why":663},"s9","Biggest jug that can measure 12 litres and 20 litres exactly","The jug size must divide both volumes; the biggest is HCF = 4 litres.",{"id":665,"label":666,"bin":625,"why":667},"s10","Bells ring every 6 and 9 minutes. When do they next ring together?","Repeating rings lining up: LCM(6, 9) = 18 minutes.","Sort everyday problems into HCF problems and LCM problems by asking: am I sharing things out, or waiting for things to meet?","This game shows ten problem cards and two bins, \"HCF problem\" and \"LCM problem\".\n\nHCF problems (cutting or sharing into the biggest equal parts): the largest square tile for a 240 cm × 180 cm floor (60 cm); the most identical boxes for 24 laddoos and 36 barfis (12 boxes); the longest equal pieces from 18 m and 24 m ribbons (6 m); the most students per row for 32 boys and 40 girls (8); the biggest jug measuring 12 L and 20 L exactly (4 L).\n\nLCM problems (repeating things meeting again, or the smallest number several numbers divide): lights flashing every 4 s and 6 s (together every 12 s); buses every 15 and 20 minutes (60 minutes); the smallest number 6 and 8 both divide (24); joggers with 8- and 12-minute laps (24 minutes); bells every 6 and 9 minutes (18 minutes).\n\nThe test: HCF answers are smaller than the given numbers; LCM answers are bigger.",{"simplerExplanation":671},"Cutting or sharing → HCF. Meeting again or smallest shared number → LCM.",{"id":673,"type":47,"variant":674,"title":675,"markdown":676},"careful-smallest","careful","Watch out for the word \"smallest\"","\"Smallest\" does not always mean HCF, and \"largest\" does not always mean LCM. \"The **smallest** number that 6 and 8 both divide\" is an **LCM** question. \"The **largest** tile\" is an **HCF** question. Think about what is happening, not just which size word appears.",{"id":678,"type":73,"title":679,"eyebrow":680,"navLabel":681},"ch09","HCF and LCM around you","Chapter 09","9 Around you",{"id":683,"type":684,"title":685,"prompt":686,"options":687},"explorer-everyday","explorer","Where the two ideas turn up","Pick a situation to see which idea it uses and why.",[688,699,710,721,732],{"id":689,"label":690,"chain":691,"badge":697,"note":698},"tiles","Floor tiles",[692,693,694,695,696],"Room 240 cm × 180 cm","Tile must fit both sides","Common factor","Largest = HCF","60 cm tiles",{"text":57,"tone":182},"Masons choose tile sizes so that whole tiles fit along the walls. A tile whose side divides both the length and the width fits without cutting; the biggest such tile is the HCF. In real homes masons also leave small gaps for grout and cut edge tiles, so the maths is the ideal version.",{"id":700,"label":701,"chain":702,"badge":708,"note":709},"sweets","Sweet boxes",[703,704,705,706,707],"24 laddoos, 36 barfis","Same mix in every box","Boxes divide both","Most boxes = HCF","12 boxes of 2 + 3",{"text":57,"tone":182},"Whenever a shopkeeper, teacher or family wants identical packets or teams from different kinds of item, the number of packets must be a common factor. Kirana shops, school events and wedding return gifts all run into this.",{"id":711,"label":712,"chain":713,"badge":719,"note":720},"buses","Bus timetables",[714,715,716,717,718],"Bus A every 15 min","Bus B every 20 min","Both leave at 8:00","Next together = LCM","9:00 a.m.",{"text":61,"tone":182},"Any two services that repeat on a fixed schedule line up again after the LCM of their gaps. Railway and bus planners use this idea to arrange connections so passengers can change easily.",{"id":722,"label":723,"chain":724,"badge":730,"note":731},"lights","Traffic lights",[725,726,727,728,729],"Signal A: 60 s cycle","Signal B: 90 s cycle","Both green at once","Repeat after LCM","every 180 s",{"text":61,"tone":182},"If one junction repeats every 60 seconds and the next every 90 seconds, the two patterns repeat together every LCM(60, 90) = 180 seconds, which is 3 minutes. Cities often give neighbouring signals the same cycle length so that a \"green wave\" can let traffic flow smoothly.",{"id":733,"label":734,"chain":735,"badge":741,"note":742},"festival","Festivals and cycles",[736,737,738,739,740],"Event A every 4 years","Event B every 6 years","Both this year","Together again after","12 years",{"text":61,"tone":182},"The Summer Olympics come every 4 years. If a family reunion happened every 6 years and both fell in the same year, they would coincide again after LCM(4, 6) = 12 years. The Extend layer looks at real calendar and sky cycles, and why nature is not always this neat.",{"id":744,"type":200,"component":745,"componentVersion":5,"config":746,"objective":762,"textAlternative":763},"lab-match-discover","match-pairs",{"prompt":747,"mode":748,"pairs":749},"Flip the cards and match each question to its answer.","memory",[750,752,754,756,758,760],{"a":751,"b":129},"HCF of 12 and 18",{"a":753,"b":127},"LCM of 4 and 6",{"a":755,"b":181},"HCF of 9 and 14",{"a":757,"b":131},"LCM of 3 and 5",{"a":759,"b":123},"HCF of 16 and 24",{"a":761,"b":134},"LCM of 6 and 9","Match small HCF and LCM questions to their answers from memory.","This memory game has twelve face-down cards: six questions and six answers. Turn two over at a time; keep them if they match.\n\nThe pairs are: HCF of 12 and 18 → **6**; LCM of 4 and 6 → **12**; HCF of 9 and 14 → **1** (they share no factor except 1); LCM of 3 and 5 → **15**; HCF of 16 and 24 → **8**; LCM of 6 and 9 → **18**.\n\nTip: an HCF answer is never bigger than the smaller number, and an LCM answer is never smaller than the bigger number. That alone rules out many wrong matches.",{"id":765,"type":47,"variant":766,"title":767,"markdown":768},"model-limit-discover","model_limit","Real life is messier than the puzzles","The timetables, shop stocks and room sizes in these puzzles are invented to give tidy numbers; none of them is a real published schedule. The puzzles also assume perfect conditions: every bus exactly on time, tiles with no gap for cement, ribbons cut with no waste, lights that never drift. Real buses run late, real tiles need grout and real clocks drift. The HCF and LCM give the ideal answer that planners start from; then they adjust for the real world.",{"id":770,"type":47,"variant":294,"title":771,"markdown":772},"try-it-calendar","Find HCF and LCM on a calendar","Take this month's calendar and a pencil.\n\n1. Circle every 3rd date (3, 6, 9, …) in one colour and every 4th date (4, 8, 12, …) in another.\n2. Which dates have both colours? You should find 12 and 24: the common multiples of 3 and 4. The first is the LCM, 12.\n3. Now look at the dates 12 and 18. Which numbers divide both of them exactly? (1, 2, 3 and 6.) The biggest, 6, is their HCF.\n\nA calendar is just a number line folded into rows of 7, so it is a handy place to spot multiples.",{"id":774,"type":151,"itemId":775,"prompt":776,"check":777,"hints":778,"feedback":781},"practice-both-hcf-lcm","hcf-and-lcm.discover-hcf-lcm-10-15","For **10** and **15**, add the HCF and the LCM together. What do you get?",{"kind":155,"answer":18,"tolerance":157},[779,780],"HCF(10, 15): the biggest number dividing both.","LCM(10, 15): the first number in both times tables.",{"correct":782,"incorrect":783},"Yes: HCF = 5 and LCM = 30, so the total is 35.","Common factors of 10 and 15 are 1 and 5, so HCF = 5. Multiples of 15: 15, 30; 30 = 10 × 3, so LCM = 30. 5 + 30 = 35.",{"id":785,"type":73,"title":786,"eyebrow":787,"navLabel":788},"ch10","More sharing and meeting problems","Chapter 10","10 More problems",{"id":790,"type":43,"markdown":791},"more-problems-intro","The best way to get comfortable is to meet lots of problems. For each one below, first decide: **sharing (HCF) or meeting (LCM)?** Then find the answer by listing, and finally do the sense check: an HCF answer is small, an LCM answer is big.",{"id":793,"type":253,"title":794,"problem":795,"steps":796},"we-diyas","Diyas in rows for Diwali","For Diwali, Ananya has **45 red diyas** and **60 yellow diyas**. She wants to arrange them in rows on the steps so that every row has the **same number** of diyas and each row is only one colour. What is the **greatest** number of diyas she can put in each row, and how many rows will there be?",[797,798,799,800,801],"Sharing into equal rows, as big as possible: this is an **HCF** problem.","Factors of 45: 1, 3, 5, 9, 15, 45.","Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.","Common factors: 1, 3, 5, 15. The greatest is **15**.","Rows: 45 ÷ 15 = 3 red rows and 60 ÷ 15 = 4 yellow rows, so **7 rows** in all.",{"id":803,"type":253,"title":804,"problem":805,"steps":806},"we-dhol","Dhol and cymbals in a Ganpati procession","In a Ganpati procession, the big dhol booms every **3 seconds** and the cymbals clash every **5 seconds**. They start together. After how many seconds do they sound together again? How many times do they sound together in the first minute (not counting the start)?",[807,808,809,810],"Two repeating sounds meeting again: this is an **LCM** problem.","Dhol: 3, 6, 9, 12, **15**, … Cymbals: 5, 10, **15**, …","LCM(3, 5) = **15 seconds**. (3 and 5 share no factor except 1, so the LCM is 3 × 5.)","In 60 seconds they sound together at 15, 30, 45 and 60 seconds: **4 times**.",{"id":812,"type":253,"title":813,"problem":814,"steps":815},"we-overs","An over and a graphic","In a cricket match, each over has **6 balls**. The TV channel shows a speed graphic every **4 balls**. Counting from the first ball of the match, after how many balls does the end of an over happen at the same moment as a graphic, for the first time?",[816,817,818,819],"Two repeating events (end of an over, a graphic) lining up: **LCM**.","Over ends: after 6, 12, 18, … balls. Graphics: after 4, 8, 12, … balls.","First shared number: LCM(6, 4) = **12 balls**, the end of the 2nd over.","Not 24 balls: 6 × 4 = 24 is a later meeting, not the first.",{"id":821,"type":47,"variant":345,"title":822,"markdown":823},"misconception-first-common","\"The HCF of 24 and 36 is 2, because 2 is the first common factor\"","Many learners find the **first** number (after 1) that divides both and stop. But the HCF is the **highest** common factor, not the first one found. The common factors of 24 and 36 are 1, 2, 3, 4, 6, 12, so HCF(24, 36) = **12**.\n\nFix: always list **all** the common factors, or keep going until you are sure nothing bigger works. A good check: divide both numbers by your answer. If the two results still share a factor (24 ÷ 2 = 12 and 36 ÷ 2 = 18 share 6), your answer was not the highest.",{"id":825,"type":47,"variant":345,"title":826,"markdown":827},"misconception-gave-up","\"9 and 12 have no common multiple, so the LCM is 9 × 12\"","A student lists 9, 18, 27 and 12, 24, sees no match, gives up and multiplies: 108. But the lists were simply too short. Keep going: 9, 18, 27, **36** and 12, 24, **36**. LCM(9, 12) = **36**.\n\nEvery pair of numbers has a common multiple (their product always works), so a match is guaranteed; you just may need to list further. Skip-counting the **bigger** number and testing with the smaller gets there fastest.",{"id":829,"type":151,"itemId":830,"prompt":831,"check":832,"hints":835,"feedback":838},"practice-taps","hcf-and-lcm.discover-dripping-taps","Two leaky taps drip every **6 seconds** and every **10 seconds**. They drip together now. After how many seconds do they next drip together?",{"kind":155,"answer":833,"tolerance":157,"unit":834},30,"seconds",[836,837],"Is this sharing or meeting?","Count in 10s and test each number with 6.",{"correct":839,"incorrect":840},"Yes: 10 ✗, 20 ✗, 30 ✓. LCM(6, 10) = 30 seconds.","It is a meeting problem, so find the LCM. Multiples of 10: 10, 20, 30. 30 = 6 × 5, so LCM = 30.",{"id":842,"type":151,"itemId":843,"prompt":844,"check":845,"hints":847,"feedback":849},"practice-mangoes","hcf-and-lcm.discover-mango-baskets","A farmer in Ratnagiri has **54 Alphonso mangoes** and **72 Kesar mangoes**. He fills baskets so that every basket has the same number of mangoes and only one variety. What is the **largest** number of mangoes per basket?",{"kind":155,"answer":206,"tolerance":157,"unit":846},"mangoes",[836,848],"The basket size must divide both 54 and 72.",{"correct":850,"incorrect":851},"Yes: HCF(54, 72) = 18. That makes 3 Alphonso baskets and 4 Kesar baskets.","Sharing into the biggest equal groups means HCF. Common factors of 54 and 72: 1, 2, 3, 6, 9, 18. The largest is 18.",{"id":853,"type":151,"itemId":854,"prompt":855,"check":856,"hints":867,"feedback":869},"practice-which-indian","hcf-and-lcm.discover-which-problem","Which of these is an **LCM** problem?",{"kind":566,"options":857,"correct":866},[858,860,862,864],{"id":271,"label":859},"Cutting 30 m and 45 m of cloth into the longest equal pieces for kurtas",{"id":274,"label":861},"Packing 40 samosas and 60 kachoris into the most identical plates",{"id":277,"label":863},"Finding when two autorickshaw stands send off rickshaws together, every 8 and 12 minutes",{"id":280,"label":865},"Finding the biggest mug that fills a 20 L bucket and a 30 L bucket exactly",[277],[868],"Look for repeating events that meet again.",{"correct":870,"incorrect":871},"Right. Rickshaws leaving every 8 and 12 minutes meet every LCM(8, 12) = 24 minutes. The others are all about cutting or sharing, so they need the HCF.","Options a, b and d are all about splitting into the biggest equal parts: HCF. Only c has two repeating events meeting: LCM.",{"id":873,"type":116,"caption":874,"columns":875,"rows":879},"table-quick-pairs","A quick reference for small pairs (all computed)",[876,877,57,878,61],"Pair","Common factors","First three common multiples",[880,883,886,889,893,895,898,902,906,912],[881,181,181,882,129],"2 and 3","6, 12, 18",[515,884,185,885,127],"1, 2","12, 24, 36",[887,884,185,888,140],"6 and 8","24, 48, 72",[890,891,187,892,134],"6 and 9","1, 3","18, 36, 54",[521,894,125,888,140],"1, 2, 4",[896,891,187,897,146],"9 and 12","36, 72, 108",[899,900,536,901,143],"10 and 15","1, 5","30, 60, 90",[903,894,125,904,905],"12 and 16","48, 96, 144","48",[907,908,909,910,911],"7 and 21","1, 7","7","21, 42, 63","21",[913,181,181,914,915],"5 and 8","40, 80, 120","40",{"id":917,"type":73,"title":918,"eyebrow":919,"navLabel":920},"ch11","Check yourself and look back","Chapter 11","11 Wrap-up",{"id":922,"type":923,"title":924,"questions":925},"quiz-discover","quiz","Sharing and meeting: quick check",[926,935,947,959,969,978,988,1001,1012,1022],{"itemId":927,"prompt":928,"options":929,"correct":277,"why":934},"hcf-and-lcm.discover-q-factor","Which of these is **not** a factor of 18?",[930,931,932,933],{"id":271,"label":185},{"id":274,"label":187},{"id":277,"label":125},{"id":280,"label":148},"18 ÷ 4 = 4 remainder 2, so 4 is not a factor. The factors of 18 are 1, 2, 3, 6, 9 and 18.",{"itemId":936,"prompt":937,"options":938,"correct":277,"why":946},"hcf-and-lcm.discover-q-multiple","Which number is a multiple of **7**?",[939,941,943,944],{"id":271,"label":940},"17",{"id":274,"label":942},"27",{"id":277,"label":525},{"id":280,"label":945},"47","35 = 7 × 5. The others leave remainders when divided by 7.",{"itemId":948,"prompt":949,"options":950,"correct":271,"why":958},"hcf-and-lcm.discover-q-common-factors","What are the common factors of **8** and **12**?",[951,952,954,956],{"id":271,"label":894},{"id":274,"label":953},"1, 2, 3, 4",{"id":277,"label":955},"2, 4, 8",{"id":280,"label":957},"1, 4","Factors of 8: 1, 2, 4, 8. Factors of 12: 1, 2, 3, 4, 6, 12. Shared: 1, 2, 4.",{"itemId":960,"prompt":961,"options":962,"correct":274,"why":968},"hcf-and-lcm.discover-q-hcf","HCF(**15**, **25**) = ?",[963,964,965,966],{"id":271,"label":181},{"id":274,"label":536},{"id":277,"label":131},{"id":280,"label":967},"75","Common factors of 15 and 25 are 1 and 5. The highest is 5.",{"itemId":970,"prompt":971,"options":972,"correct":277,"why":977},"hcf-and-lcm.discover-q-lcm","LCM(**3**, **4**) = ?",[973,974,975,976],{"id":271,"label":181},{"id":274,"label":909},{"id":277,"label":127},{"id":280,"label":140},"Multiples of 4: 4, 8, 12. 12 is also a multiple of 3, so LCM = 12.",{"itemId":979,"prompt":980,"options":981,"correct":274,"why":987},"hcf-and-lcm.discover-q-lcm-not-product","LCM(**6**, **4**) = ?",[982,983,984,985],{"id":271,"label":185},{"id":274,"label":127},{"id":277,"label":140},{"id":280,"label":986},"10","Multiples of 6: 6, 12. 12 = 4 × 3, so LCM = 12. 24 is a common multiple but not the lowest.",{"itemId":989,"prompt":990,"options":991,"correct":271,"why":1000},"hcf-and-lcm.discover-q-which-tiles","You want the **largest** square tiles for a room 300 cm × 240 cm. Which do you find?",[992,994,996,998],{"id":271,"label":993},"HCF of 300 and 240",{"id":274,"label":995},"LCM of 300 and 240",{"id":277,"label":997},"300 × 240",{"id":280,"label":999},"300 − 240","The tile side must divide both lengths exactly, and you want the biggest: HCF(300, 240) = 60 cm.",{"itemId":1002,"prompt":1003,"options":1004,"correct":274,"why":1011},"hcf-and-lcm.discover-q-which-bells","Two bells ring every 10 and 15 minutes. To find when they ring together, you find the…",[1005,1006,1007,1009],{"id":271,"label":57},{"id":274,"label":61},{"id":277,"label":1008},"sum",{"id":280,"label":1010},"difference","Repeating events meet at a common multiple; the first one is LCM(10, 15) = 30 minutes.",{"itemId":1013,"prompt":1014,"options":1015,"correct":271,"why":1021},"hcf-and-lcm.discover-q-coprime","HCF(**7**, **10**) = ?",[1016,1017,1018,1019],{"id":271,"label":181},{"id":274,"label":909},{"id":277,"label":986},{"id":280,"label":1020},"70","7 and 10 share no factor except 1, so their HCF is 1. Such numbers are called co-primes.",{"itemId":1023,"prompt":1024,"options":1025,"correct":271,"why":1034},"hcf-and-lcm.discover-q-sense","Which statement is always true for two whole numbers?",[1026,1028,1030,1032],{"id":271,"label":1027},"HCF ≤ smaller number ≤ bigger number ≤ LCM",{"id":274,"label":1029},"LCM ≤ HCF",{"id":277,"label":1031},"HCF = LCM",{"id":280,"label":1033},"LCM is always the product","The HCF divides both numbers, so it is no bigger than either. Both numbers divide the LCM, so it is no smaller than either.",{"id":1036,"type":1037,"title":1038,"points":1039},"cheat-sheet-discover","summary","Cheat sheet",[1040,1041,1042,1043,1044,1045,1046,1047,1048],"**Factor:** divides a number exactly. Factors of 12: 1, 2, 3, 4, 6, 12. Find them in pairs.","**Multiple:** the number’s times table. Multiples of 4: 4, 8, 12, 16, … They never end.","**Common factors** are in both factor lists; the biggest is the **HCF** (also called GCD or GCF). HCF(12, 18) = 6.","**Common multiples** are on both multiple trails; the smallest is the **LCM**. LCM(4, 6) = 12.","**HCF problems:** cutting, sharing, packing or arranging into the **largest** equal parts. The answer is small.","**LCM problems:** things repeating and **meeting again**, or the **smallest** number several numbers divide. The answer is big.","**Sense check:** HCF ≤ each number ≤ LCM.","**Co-primes** (like 8 and 9) share only the factor 1: HCF = 1 and LCM = their product.","**Common mistake:** the LCM is not always the product. LCM(4, 6) = 12, not 24.",{"id":1050,"type":1051,"prompt":1052},"reflect-discover","reflection","Think of one thing in your home, school or town that repeats on a schedule (a bell, a bus, a festival, a medicine dose), and one thing that gets shared or cut into equal parts. Write one HCF puzzle and one LCM puzzle from them, with numbers, and solve both.",{"id":1054,"type":1055,"conceptId":1056,"relation":1057,"explanation":1058},"conn-prime","connection","prime-and-composite","helps_understand","Prime numbers are the building blocks for the fastest HCF and LCM methods; co-primes have HCF 1.",{"id":1060,"type":1055,"conceptId":1061,"relation":1062,"explanation":1063},"conn-four-ops","four-operations","related_to","Finding factors and multiples is multiplication and division: HCF and LCM depend on fluent times tables.",{"id":1065,"type":1055,"conceptId":1066,"relation":1062,"explanation":1067},"conn-patterns","patterns","Multiples form number patterns; common multiples repeat in a regular pattern every LCM.",{"id":1069,"type":1070,"sourceIds":1071},"sources-discover","sources",[1072,1073,1074,1075,1076],"hcf-and-lcm-ncert-class6-playing-with-numbers","hcf-and-lcm-mathsisfun-gcf","hcf-and-lcm-mathsisfun-lcm","hcf-and-lcm-khan-factors-multiples","hcf-and-lcm-nrich-factors-multiples-primes",[1072,1073,1074,1075,1076],"needs_review",{"generatedBy":1080,"notes":1081},"claude-code","Draft generated with Python-checked arithmetic; pending owner review.","19ca23368f9cf72326b28fc61685c5ae199c6d8f89a158d0c07f24e518898eb4",{"logic:practice":1084,"component:hcf-lcm@1":1085,"component:sort-game@1":1086,"component:match-pairs@1":1087,"source:hcf-and-lcm-khan-factors-multiples":1088,"source:hcf-and-lcm-mathsisfun-gcf":1089,"source:hcf-and-lcm-mathsisfun-lcm":1090,"source:hcf-and-lcm-ncert-class6-playing-with-numbers":1091,"source:hcf-and-lcm-nrich-factors-multiples-primes":1092},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","83f068508b17b70184e45fbcf95d356790bfa2b0a98dcdb859f8e9dc105e5d33","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","01af9d960126bcdac368f15ef1b43edbf2e389c0ce966c7e903721b346655feb","8499aa4d85d927e518cacd55fa1f7870d1da1332351d07b641c2a1cc6c885659","722e7f7db3c09e9e29bf5c0f363173f52eb56ef08d1bd84ad3f51890d1375d07","82c5fcc908046567f1a1bdb191f9b28fd42600389720d981bc1965d0c1542772","25674a9fbe152fe3c9ce7273cc44537dd3ebb9b393ed8ba01f12daf10e9d473f",{"state":1094,"reviewer":1095,"selfReview":419,"reviewedAt":1096,"method":1097},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899598618]