[{"data":1,"prerenderedAt":1046},["ShallowReactive",2],{"layer:prime-and-composite:discover":3},{"layer":4,"contentHash":1027,"dependencyHashes":1028,"approval":1039,"releaseId":1045},{"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":1022,"reviewStatus":1023,"authoring":1024},1,"prime-and-composite","en","discover","Numbers that will not make rectangles","Factors, multiples and the numbers that can only stand in a single line","Share laddoos, set out chairs and build rectangles from tiles to meet factors and multiples. Discover prime numbers, composite numbers, the odd case of 1, the Sieve of Eratosthenes, twin primes and co-primes.",[13,14,15,16,17],"Find all the factors of a number up to 50 using factor pairs, and list its first few multiples.","Explain the difference between a factor and a multiple.","Sort numbers into prime, composite or neither, and explain why 1 is neither and 2 is the only even prime.","Use the Sieve of Eratosthenes to find the primes up to 50.","Recognise twin primes and co-prime pairs.",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","20 buttons, beans or coins",{"label":32,"value":33},"Chapters","10",{"label":35,"value":36},"Labs","Rectangles, factor match, prime sort, sieve, hunt",{"label":38,"value":39},"Big idea","Primes have exactly two factors",[41,45,51,57,60,65,80,85,100,115,120,123,127,151,155,160,169,192,228,233,236,277,282,358,363,366,370,381,386,389,393,441,445,466,479,550,555,558,562,565,570,575,596,601,604,634,645,661,665,668,680,684,695,700,703,714,717,721,742,747,750,798,804,808,813,818,877,992,996,1012],{"id":42,"type":43,"markdown":44},"intro-laddoos","prose","Amma has made **12 laddoos** for Diwali and wants to pack them in a flat box, in neat rows, with every row the same length. She can do it lots of ways: **1 row of 12**, **2 rows of 6**, **3 rows of 4**, **4 rows of 3**, **6 rows of 2** or **12 rows of 1**.\n\nThe next day she makes **13 laddoos**. Try as she might, the only neat box is one long line: **1 row of 13**. Two rows? One row would have 7 and the other 6. Three rows? 4, 4 and 5. Every choice leaves a laddoo sticking out.\n\nWhy can 12 be split so many ways while 13 refuses? That one question leads to some of the oldest and most important ideas in mathematics: **factors**, **multiples**, and the special numbers called **primes**.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"how-to-use","callout","observation","How to use this lesson","Read the chapters in order the first time. Keep a pencil and some small things to arrange (buttons, coins, beads or dried rajma) close by: nearly every idea here can be built with your hands. When you see a **prediction**, choose an answer before reading on. Guessing wrong and then seeing why is one of the best ways to learn.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch1","chapter","Sharing without leftovers","Chapter 01","1 Sharing fairly",{"id":58,"type":43,"markdown":59},"sharing-prose","Think about sharing things fairly. You have **12 pencils** and want to share them equally with no pencils left over.\n\n- Between **2** friends: 6 each. Works.\n- Between **3** friends: 4 each. Works.\n- Between **5** friends: 2 each, with 2 left over. Does **not** work.\n\nWhen a number can be shared equally into groups with nothing left over, we say it **divides exactly**. 12 ÷ 3 = 4 exactly, but 12 ÷ 5 = 2 with a remainder of 2.\n\nThe numbers that divide 12 exactly are **1, 2, 3, 4, 6, 12**. These are called the **factors** of 12. You can check each one: 12 ÷ 1 = 12, 12 ÷ 2 = 6, 12 ÷ 3 = 4, 12 ÷ 4 = 3, 12 ÷ 6 = 2 and 12 ÷ 12 = 1. No remainders anywhere.",{"id":61,"type":47,"variant":62,"title":63,"markdown":64},"def-factor","definition","Factor","A **factor** of a number divides it exactly, leaving **no remainder**.\n\nThe factors of 12 are 1, 2, 3, 4, 6, 12. The factors of 13 are only 1, 13. The factors of 20 are 1, 2, 4, 5, 10, 20.",{"id":66,"type":67,"title":68,"problem":69,"steps":70,"help":78},"we-factors-18","worked_example","All the factors of 18","Find every factor of 18.",[71,72,73,74,75,76,77],"Start with 1. Every number can be shared into 1 group: 18 ÷ 1 = 18. So **1** and **18** are factors.","Try 2: 18 ÷ 2 = 9 exactly. So **2** and **9** are factors.","Try 3: 18 ÷ 3 = 6 exactly. So **3** and **6** are factors.","Try 4: 18 ÷ 4 = 4 remainder 2. Not a factor.","Try 5: 18 ÷ 5 = 3 remainder 3. Not a factor.","The next number to try is 6, but we already have it (from 3 × 6). The factors have started repeating, so we can stop.","The factors of 18 are **1, 2, 3, 6, 9, 18**. There are 6 of them.",{"simplerExplanation":79},"Try dividing 18 by 1, 2, 3 and so on. Each time there is no remainder, you have found two factors at once: the number you divided by and the answer.",{"id":81,"type":47,"variant":82,"title":83,"markdown":84},"aha-pairs","aha","Factors come in pairs","Look at the worked example again. Every time we found one factor, we got a partner for free: 2 came with 9, and 3 came with 6. That is because **2 × 9 = 18** and **3 × 6 = 18**.\n\nThese partners are called **factor pairs**. For 36 the pairs are 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6. Notice the last pair: **6 × 6**. The partner is the number itself, so 36 has an odd number of factors (9 of them).",{"id":86,"type":87,"itemId":88,"prompt":89,"check":90,"hints":94,"feedback":97},"prac-factors-20","practice","prime-and-composite.discover-count-factors-20","How many factors does **20** have? (Count 1 and 20 too.)",{"kind":91,"answer":92,"tolerance":93},"number",6,0,[95,96],"Look for pairs that multiply to 20: 1 × 20, 2 × 10, …","Does 3 divide 20 exactly? Does 4?",{"correct":98,"incorrect":99},"Yes: 1, 2, 4, 5, 10, 20. That is 6 factors, in 3 pairs: 1 × 20, 2 × 10, 4 × 5.","The pairs are 1 × 20, 2 × 10 and 4 × 5, so the factors are 1, 2, 4, 5, 10, 20: 6 in all. 3 does not work (20 ÷ 3 leaves 2).",{"id":101,"type":102,"prompt":103,"options":104,"explanation":114},"pred-one-factor","prediction","Is **1** a factor of every counting number?",[105,108,111],{"id":106,"label":107},"a","Yes, always",{"id":109,"label":110},"b","Only of odd numbers",{"id":112,"label":113},"c","Only of 1 itself","**Yes, always.** Any number of things can be put into **1** group with nothing left over: 57 ÷ 1 = 57, 1,000 ÷ 1 = 1,000. So 1 is a factor of every counting number, and every number is a factor of itself. These two factors, 1 and the number, are sometimes called the **trivial** or obvious factors. The interesting question is always: are there any others?",{"id":116,"type":53,"title":117,"eyebrow":118,"navLabel":119},"ch2","Multiples: the times tables go on forever","Chapter 02","2 Multiples",{"id":121,"type":43,"markdown":122},"multiples-prose","Now turn the idea around. Start with 4 and keep adding 4: **4, 8, 12, 16, 20, 24, 28, …** These are the **multiples** of 4. They are just the 4 times table: 4 × 1, 4 × 2, 4 × 3 and so on.\n\nMultiples never stop. Whatever multiple you reach, you can always add 4 again. The 4 times table has no last number.\n\nFactors are different. A number has **only a few factors**, and none of them is bigger than the number itself. 12 has exactly six factors and that is the end of the list.",{"id":124,"type":47,"variant":62,"title":125,"markdown":126},"def-multiple","Multiple","A **multiple** of a number is what you get when you multiply it by a counting number (1, 2, 3, …).\n\nMultiples of 4: 4, 8, 12, 16, 20, … Multiples of 7: 7, 14, 21, 28, 35, …\n\nThe first multiple of every number is the number itself (4 × 1 = 4).",{"id":128,"type":129,"tone":130,"items":131},"spec-fm","spec","blue",[132,136,140,144,148],{"label":133,"big":134,"value":135},"Factors of 12","6 of them","1, 2, 3, 4, 6, 12. A short list that ends. None is bigger than 12.",{"label":137,"big":138,"value":139},"Multiples of 12","forever","12, 24, 36, 48, 60, … The list never ends. None is smaller than 12.",{"label":141,"big":142,"value":143},"Smallest factor","1","Every number has 1 as a factor.",{"label":145,"big":146,"value":147},"Largest factor","itself","Every number is a factor of itself.",{"label":149,"big":146,"value":150},"Smallest multiple","Every number is its own first multiple.",{"id":152,"type":47,"variant":82,"title":153,"markdown":154},"aha-link","Two ways of saying the same thing","**3 is a factor of 12** and **12 is a multiple of 3** describe exactly the same fact: 3 × 4 = 12.\n\nSo every number is a multiple of each of its factors. 12 is a multiple of 1, 2, 3, 4, 6 and 12. If you know one, you know the other.",{"id":156,"type":47,"variant":157,"title":158,"markdown":159},"misc-fm","misconception","“Factors are big, multiples are small”","It is the other way round, and it is easy to mix up. **Factors fit into** a number, so they are the same size or smaller. **Multiples are built from** a number, so they are the same size or bigger.\n\nA memory trick: *Factors are Few and Fit inside; Multiples are Many and More.*",{"id":161,"type":67,"title":162,"problem":163,"steps":164},"we-multiples-table","Spotting factors and multiples on a table","Use the 6 times table, 6, 12, 18, 24, 30, 36, to answer: is 6 a factor of 30? Is 36 a multiple of 6? Is 6 a factor of 40?",[165,166,167,168],"30 is in the 6 times table (6 × 5 = 30). So 30 is a multiple of 6, and **6 is a factor of 30**.","36 is in the table (6 × 6 = 36), so **36 is a multiple of 6**.","40 is not in the table: it sits between 36 and 42. So 40 ÷ 6 leaves a remainder of 4, and **6 is not a factor of 40**.","One table answers both kinds of question, because factor and multiple are two views of the same multiplication.",{"id":170,"type":87,"itemId":171,"prompt":172,"check":173,"hints":186,"feedback":189},"prac-multiple-of-6","prime-and-composite.discover-multiple-of-6","Which of these is a **multiple of 6**?",{"kind":174,"options":175,"correct":185},"choice",[176,178,180,182],{"id":106,"label":177},"3",{"id":109,"label":179},"16",{"id":112,"label":181},"42",{"id":183,"label":184},"d","62",[112],[187,188],"A multiple of 6 is in the 6 times table.","Divide each number by 6. Which leaves no remainder?",{"correct":190,"incorrect":191},"42 = 6 × 7, so 42 is in the 6 times table.","Only 42 works: 42 = 6 × 7. 3 is a factor of 6, not a multiple. 16 ÷ 6 and 62 ÷ 6 both leave remainders.",{"id":193,"type":194,"component":195,"componentVersion":5,"config":196,"objective":222,"textAlternative":223,"help":224},"lab-factor-match","interactive","match-pairs",{"prompt":197,"mode":198,"pairs":199},"Match each number to the complete list of its factors.","memory",[200,203,206,208,211,214,216,219],{"a":201,"b":202},"6","1, 2, 3, 6",{"a":204,"b":205},"9","1, 3, 9",{"a":33,"b":207},"1, 2, 5, 10",{"a":209,"b":210},"13","1, 13",{"a":212,"b":213},"15","1, 3, 5, 15",{"a":179,"b":215},"1, 2, 4, 8, 16",{"a":217,"b":218},"18","1, 2, 3, 6, 9, 18",{"a":220,"b":221},"24","1, 2, 3, 4, 6, 8, 12, 24","Flip cards to match a number with the full list of its factors.","This memory game hides 16 cards: 8 numbers and 8 factor lists. Turn over two at a time and keep them if they match.\n\n- **6** → 1, 2, 3, 6\n- **9** → 1, 3, 9\n- **10** → 1, 2, 5, 10\n- **13** → 1, 13\n- **15** → 1, 3, 5, 15\n- **16** → 1, 2, 4, 8, 16\n- **18** → 1, 2, 3, 6, 9, 18\n- **24** → 1, 2, 3, 4, 6, 8, 12, 24\n\n13 is the only number here with just two factors, so its card is the shortest: 13 is prime. 9 and 16 are square numbers, and they are the only ones with an odd number of factors.",{"hints":225},[226,227],"The list always starts with 1 and ends with the number itself.","Short lists belong to primes. 13 has the shortest.",{"id":229,"type":53,"title":230,"eyebrow":231,"navLabel":232},"ch3","The rectangle picture","Chapter 03","3 Rectangles",{"id":234,"type":43,"markdown":235},"rect-prose","Here is a picture you can carry in your head. Take a number of square tiles and try to make **rectangles** from all of them, with no gaps and no tiles left over.\n\nWith **12 tiles** you can make a 1 by 12 strip, a 2 by 6 rectangle or a 3 by 4 rectangle. (A 4 by 3 is the same rectangle turned on its side.) The side lengths of the rectangles are exactly the factor pairs of 12: 1 × 12, 2 × 6, 3 × 4.\n\nWith **7 tiles** you can only make a 1 by 7 strip. Any other shape leaves a tile hanging off the edge.\n\nRangoli makers, gardeners planting rows of saplings and teachers setting out chairs for a school assembly all use this idea. The number of rows and the number in each row must be a factor pair of the total.",{"id":237,"type":238,"caption":239,"columns":240,"rows":245},"table-rects","table","Rectangles you can make with n tiles (a 2 × 6 and a 6 × 2 count as the same rectangle)",[241,242,243,244],"Tiles","Rectangles (rows × columns)","Number of rectangles","Only a strip?",[246,250,254,257,259,261,264,267,269,271,274],[201,247,248,249],"1 × 6, 2 × 3","2","No",[251,252,142,253],"7","1 × 7","Yes",[255,256,248,249],"8","1 × 8, 2 × 4",[204,258,248,249],"1 × 9, 3 × 3",[33,260,248,249],"1 × 10, 2 × 5",[262,263,142,253],"11","1 × 11",[265,266,177,249],"12","1 × 12, 2 × 6, 3 × 4",[209,268,142,253],"1 × 13",[179,270,177,249],"1 × 16, 2 × 8, 4 × 4",[272,273,142,253],"17","1 × 17",[220,275,276,249],"1 × 24, 2 × 12, 3 × 8, 4 × 6","4",{"id":278,"type":47,"variant":279,"title":280,"markdown":281},"tryit-tiles","try_it","Build it with buttons","Collect 20 buttons, coins or dried beans.\n\n1. Count out 10 and make every rectangle you can. Write down rows × columns each time.\n2. Do the same for 11. What goes wrong?\n3. Now try 16. One of its rectangles is special: it is a **square**. Which one?\n4. Try every number from 2 to 20. Circle the numbers that only make a single strip.\n\nKeep your list. By the end of this lesson you will know what to call the circled numbers.",{"id":283,"type":194,"component":284,"componentVersion":5,"config":285,"objective":351,"textAlternative":352,"help":353},"lab-rect-sort","sort-game",{"prompt":286,"bins":287,"items":294,"seconds":93},"Can this many things be set out as a rectangle with at least 2 rows and at least 2 in each row?",[288,291],{"id":289,"label":290},"rect","Makes a rectangle",{"id":292,"label":293},"strip","Only a single line",[295,299,303,307,311,315,319,323,327,331,335,339,343,347],{"id":296,"label":297,"bin":289,"why":298},"n6","6 marigold pots","6 = 2 × 3, so 2 rows of 3 pots work.",{"id":300,"label":301,"bin":292,"why":302},"n7","7 chairs","7 can only be 1 × 7. Any other rows leave a chair over.",{"id":304,"label":305,"bin":289,"why":306},"n9","9 saplings","9 = 3 × 3: a square of 3 rows of 3.",{"id":308,"label":309,"bin":292,"why":310},"n11","11 cricket players","11 is only 1 × 11. That is why a team lines up in a single row for the anthem.",{"id":312,"label":313,"bin":289,"why":314},"n12","12 laddoos","12 = 2 × 6 or 3 × 4.",{"id":316,"label":317,"bin":292,"why":318},"n13","13 laddoos","13 is only 1 × 13.",{"id":320,"label":321,"bin":289,"why":322},"n15","15 diyas","15 = 3 × 5.",{"id":324,"label":325,"bin":292,"why":326},"n17","17 bangles","17 is only 1 × 17.",{"id":328,"label":329,"bin":289,"why":330},"n21","21 students","21 = 3 × 7: three rows of seven.",{"id":332,"label":333,"bin":292,"why":334},"n23","23 mangoes","23 is only 1 × 23. 23 is not in any times table except 1 and 23.",{"id":336,"label":337,"bin":289,"why":338},"n25","25 tiles","25 = 5 × 5, a square.",{"id":340,"label":341,"bin":292,"why":342},"n29","29 stamps","29 is only 1 × 29.",{"id":344,"label":345,"bin":292,"why":346},"n2","2 shoes","2 can only be 1 × 2: you cannot have 2 rows with at least 2 in each.",{"id":348,"label":349,"bin":289,"why":350},"n27","27 marbles","27 = 3 × 9. Many people guess 27 is a strip number, but 3 × 9 = 27.","Decide whether a number of objects can be arranged in a rectangle of at least 2 rows, or only in one line.","This game shows 14 cards, each with a number of objects, and two bins: **Makes a rectangle** (at least 2 rows with at least 2 in each row) and **Only a single line**.\n\n- Rectangle numbers: 6 (2 × 3), 9 (3 × 3), 12 (2 × 6 or 3 × 4), 15 (3 × 5), 21 (3 × 7), 25 (5 × 5) and 27 (3 × 9).\n- Single-line numbers: 2, 7, 11, 13, 17, 23 and 29. The only way to set these out is 1 row.\n\nThe trick card is 27. It is odd and does not look like a times-table answer, but 27 = 3 × 9. To decide, try splitting the objects into 2 rows, then 3 rows, then 4 rows and so on. If one of them works with nothing left over, it is a rectangle number.",{"simplerExplanation":354,"hints":355},"Try to put the objects in 2 equal rows. If that fails, try 3 equal rows, then 4. If nothing works except one long row, it goes in the single-line bin.",[356,357],"Even numbers bigger than 2 can always make 2 rows.","Is the number in the 3 times table? Then 3 rows will work.",{"id":359,"type":53,"title":360,"eyebrow":361,"navLabel":362},"ch4","Prime numbers: only two factors","Chapter 04","4 Prime numbers",{"id":364,"type":43,"markdown":365},"prime-prose","The numbers that can only make a single strip have a special name. They are called **prime numbers**.\n\nA prime number has **exactly two factors**: **1** and **the number itself**. 7 has factors 1 and 7. 13 has factors 1 and 13. 29 has factors 1 and 29. Nothing else divides them exactly.\n\nThe first few prime numbers are **2, 3, 5, 7, 11, 13, 17, 19, 23, 29**.\n\nThe word *prime* comes from the Latin *primus*, meaning *first*. Prime numbers are \"first\" in the sense that they are the basic building blocks: every other counting number bigger than 1 can be made by multiplying primes together, as you will see in later layers.",{"id":367,"type":47,"variant":62,"title":368,"markdown":369},"def-prime","Prime number","A **prime number** is a counting number that has **exactly two different factors**: 1 and itself.\n\nExamples: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.",{"id":371,"type":102,"prompt":372,"options":373,"explanation":380},"pred-prime-51","Is **51** a prime number?",[374,376,378],{"id":106,"label":375},"Yes: it is odd and does not end in 5",{"id":109,"label":377},"No: it has more than two factors",{"id":112,"label":379},"Cannot tell without a calculator","**No.** 51 = 3 × 17, so its factors are 1, 3, 17 and 51. That is four factors, not two.\n\n51 fools many people because it is odd and not in the 5 times table. A quick check: add the digits, 5 + 1 = 6. Because 6 is in the 3 times table, 51 is too. (You will learn why that trick works in a later layer.) Other famous fakes are 57 = 3 × 19, 87 = 3 × 29 and 91 = 7 × 13.",{"id":382,"type":53,"title":383,"eyebrow":384,"navLabel":385},"ch5","Composite numbers: more than two factors","Chapter 05","5 Composite numbers",{"id":387,"type":43,"markdown":388},"comp-prose","Numbers like 12, 15, 21 and 27 can make rectangles because they have **more than two factors**. They are called **composite numbers**.\n\n*Composite* means \"made of several parts\". A composite number can be built by multiplying two smaller numbers (both bigger than 1): 12 = 3 × 4, 15 = 3 × 5, 27 = 3 × 9.\n\nEvery even number bigger than 2 is composite, because it can always be split into 2 equal rows. For example 14 = 2 × 7 and 100 = 2 × 50.",{"id":390,"type":47,"variant":62,"title":391,"markdown":392},"def-composite","Composite number","A **composite number** is a counting number that has **more than two factors**.\n\nExamples: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21. The smallest composite number is **4** (factors 1, 2, 4).",{"id":394,"type":238,"caption":395,"columns":396,"rows":401},"table-1to20","The numbers 1 to 20 sorted by how many factors they have",[397,398,399,400],"Number","Factors","How many","Type",[402,404,407,409,412,415,416,418,420,421,422,424,426,427,430,431,432,434,435,438],[142,142,142,403],"neither",[248,405,248,406],"1, 2","prime",[177,408,248,406],"1, 3",[276,410,177,411],"1, 2, 4","composite",[413,414,248,406],"5","1, 5",[201,202,276,411],[251,417,248,406],"1, 7",[255,419,276,411],"1, 2, 4, 8",[204,205,177,411],[33,207,276,411],[262,423,248,406],"1, 11",[265,425,201,411],"1, 2, 3, 4, 6, 12",[209,210,248,406],[428,429,276,411],"14","1, 2, 7, 14",[212,213,276,411],[179,215,413,411],[272,433,248,406],"1, 17",[217,218,201,411],[436,437,248,406],"19","1, 19",[439,440,201,411],"20","1, 2, 4, 5, 10, 20",{"id":442,"type":47,"variant":48,"title":443,"markdown":444},"obs-table","What the table shows","Between 1 and 20 there are **8 primes** (2, 3, 5, 7, 11, 13, 17, 19), **11 composites**, and one number, **1**, that is neither. The number with the most factors is **12**, together with 18 and 20: each has six factors.",{"id":446,"type":87,"itemId":447,"prompt":448,"check":449,"hints":460,"feedback":463},"prac-which-prime","prime-and-composite.discover-which-prime","Which **one** of these numbers is prime?",{"kind":174,"options":450,"correct":459},[451,453,455,457],{"id":106,"label":452},"21",{"id":109,"label":454},"27",{"id":112,"label":456},"31",{"id":183,"label":458},"35",[112],[461,462],"Try to divide each number by 3, 5 and 7.","21 = 3 × 7. What about the others?",{"correct":464,"incorrect":465},"31 has only the factors 1 and 31. The others are 21 = 3 × 7, 27 = 3 × 9 and 35 = 5 × 7.","The prime is 31. The others each have more than two factors: 21 = 3 × 7, 27 = 3 × 9, 35 = 5 × 7.",{"id":467,"type":67,"title":468,"problem":469,"steps":470,"help":477},"we-is-37-prime","Is 37 prime?","A class has 37 students. Can the teacher arrange them in equal rows (more than one row, more than one student in each)? In other words, is 37 prime?",[471,472,473,474,475,476],"Try 2 rows: 37 is odd, so 37 ÷ 2 leaves 1 over. No.","Try 3 rows: 3 × 12 = 36, so 37 ÷ 3 leaves 1 over. No.","Try 4 rows: 4 × 9 = 36, so 1 over again. No. (If 2 does not work, 4 cannot work either.)","Try 5 rows: 37 does not end in 0 or 5. No.","Try 6 rows: 6 × 6 = 36. No. Now 6 × 6 is already very close to 37, and 7 × 7 = 49 is too big, so if a pair existed we would have found its smaller number by now.","Only 1 row of 37 works. **37 is prime.** Its factors are just 1 and 37.",{"simplerExplanation":478},"Keep trying to share 37 into 2, 3, 4, 5 and 6 equal rows. Every try leaves some over, so 37 cannot make a rectangle: it is prime.",{"id":480,"type":194,"component":284,"componentVersion":5,"config":481,"objective":544,"textAlternative":545,"help":546},"lab-pcn-sort",{"prompt":482,"bins":483,"items":490,"seconds":93},"Sort each number: prime, composite, or neither?",[484,486,488],{"id":406,"label":485},"Prime",{"id":411,"label":487},"Composite",{"id":403,"label":489},"Neither",[491,494,497,500,503,506,510,513,516,520,524,528,532,536,540],{"id":492,"label":142,"bin":403,"why":493},"c1","1 has only one factor, itself.",{"id":495,"label":248,"bin":406,"why":496},"c2","Factors 1 and 2 only. The only even prime.",{"id":498,"label":276,"bin":411,"why":499},"c4","1, 2, 4: three factors. The smallest composite.",{"id":501,"label":413,"bin":406,"why":502},"c5","Factors 1 and 5 only.",{"id":504,"label":204,"bin":411,"why":505},"c9","9 = 3 × 3. Odd, but composite.",{"id":507,"label":508,"bin":406,"why":509},"c11","11 (players in a cricket team)","11 has only the factors 1 and 11.",{"id":511,"label":428,"bin":411,"why":512},"c14","14 = 2 × 7.",{"id":514,"label":436,"bin":406,"why":515},"c19","Factors 1 and 19 only.",{"id":517,"label":518,"bin":411,"why":519},"c24","24 (hours in a day)","24 has eight factors: 1, 2, 3, 4, 6, 8, 12, 24.",{"id":521,"label":522,"bin":406,"why":523},"c31","31 (days in May)","31 has only the factors 1 and 31.",{"id":525,"label":526,"bin":411,"why":527},"c33","33","33 = 3 × 11.",{"id":529,"label":530,"bin":411,"why":531},"c39","39","39 = 3 × 13. A common trap.",{"id":533,"label":534,"bin":406,"why":535},"c41","41","41 has only the factors 1 and 41.",{"id":537,"label":538,"bin":411,"why":539},"c49","49","49 = 7 × 7.",{"id":541,"label":542,"bin":411,"why":543},"c60","60 (minutes in an hour)","60 has twelve factors, which is why an hour splits so neatly into halves, thirds, quarters and more.","Sort numbers into prime, composite or neither by thinking about how many factors they have.","This game has 15 number cards and three bins: **Prime**, **Composite** and **Neither**.\n\n- **Prime** (exactly two factors): 2, 5, 11, 19, 31 and 41.\n- **Composite** (more than two factors): 4, 9 (3 × 3), 14 (2 × 7), 24, 33 (3 × 11), 39 (3 × 13), 49 (7 × 7) and 60.\n- **Neither:** 1, because it has only one factor.\n\nThe traps are the odd composites 9, 33, 39 and 49, which look prime at first glance. Some cards have real-life labels: 11 players in a cricket team (prime), 24 hours in a day and 60 minutes in an hour (both composite, with lots of factors, which is why days and hours split so neatly).",{"hints":547},[548,549],"Look for a way to split the number into two smaller whole numbers multiplied together.","Remember 1 has its own bin.",{"id":551,"type":53,"title":552,"eyebrow":553,"navLabel":554},"ch6","Two special numbers: 1 and 2","Chapter 06","6 The numbers 1 and 2",{"id":556,"type":43,"markdown":557},"one-prose","Look back at the table. The number **1** has only **one** factor: itself. A prime needs exactly two different factors, and a composite needs more than two. So 1 is **neither prime nor composite**. It is in a group of its own.\n\nMathematicians also have a deeper reason for leaving 1 out of the primes. Primes are the building blocks that other numbers are made from, and each number has **only one recipe** of primes (12 = 2 × 2 × 3, and no other set of primes multiplies to 12). If 1 counted as a prime, you could write 12 = 2 × 2 × 3 × 1 × 1 × 1 and the recipe would stop being unique. You will meet this idea properly in the Deepen layer.",{"id":559,"type":47,"variant":157,"title":560,"markdown":561},"misc-one","“1 is the first prime number”","Many people think this, and until about a hundred years ago some mathematicians wrote it too. Today everyone agrees: **1 is not prime** and **1 is not composite**. It has only one factor. The first prime number is **2**.",{"id":563,"type":43,"markdown":564},"two-prose","The number **2** is special in another way. It is the **only even prime number**.\n\nEvery even number can be split into 2 equal groups, so 2 is a factor of every even number. For 2 itself that is fine: its factors are just 1 and 2. But any bigger even number, such as 4, 6, 8 or 100, has at least three factors: 1, 2 and itself. So every even number after 2 is composite.\n\nThat means that, apart from 2, **all primes are odd**. But be careful: not all odd numbers are prime. 9, 15, 21, 25 and 27 are odd and composite.",{"id":566,"type":47,"variant":567,"title":568,"markdown":569},"careful-odd","careful","Odd does not mean prime","Odd numbers can be composite, and 2 is a prime that is even. The quickest mistake in this topic is to say \"it is odd, so it is prime\". Always look for factors. 9 = 3 × 3, 25 = 5 × 5, 49 = 7 × 7 and 91 = 7 × 13 are all odd composites.",{"id":571,"type":47,"variant":572,"title":573,"markdown":574},"ex-one-tile","example","The rectangle picture for 1 and 2","With **1 tile** you get a single square, 1 × 1. There is only one side length, 1, so there is only one factor.\n\nWith **2 tiles** you get a 1 × 2 strip: two different factors, 1 and 2, so 2 is prime.\n\nWith **4 tiles** you get a 1 × 4 strip **and** a 2 × 2 square: factors 1, 2 and 4, so 4 is composite.",{"id":576,"type":87,"itemId":577,"prompt":578,"check":579,"hints":590,"feedback":593},"prac-true-false","prime-and-composite.discover-true-statement","Which statement is **true**?",{"kind":174,"options":580,"correct":589},[581,583,585,587],{"id":106,"label":582},"1 is the smallest prime number.",{"id":109,"label":584},"All odd numbers are prime.",{"id":112,"label":586},"2 is the only even prime number.",{"id":183,"label":588},"9 is prime because it is odd.",[112],[591,592],"Think about what factors 1 has.","Is 9 = 3 × 3?",{"correct":594,"incorrect":595},"Right. Every even number bigger than 2 has 2 as an extra factor, so 2 is the only even prime.","Only (c) is true. 1 is neither prime nor composite, and 9 = 3 × 3 shows odd numbers can be composite.",{"id":597,"type":53,"title":598,"eyebrow":599,"navLabel":600},"ch7","The Sieve of Eratosthenes","Chapter 07","7 The Sieve",{"id":602,"type":43,"markdown":603},"sieve-intro","How can you find all the primes up to 50 without testing every number one at a time? More than 2,200 years ago, a Greek scholar named **Eratosthenes**, the chief librarian at Alexandria in Egypt, described a clever shortcut. It is called the **Sieve of Eratosthenes**.\n\nA sieve is like the chalni in the kitchen that lets fine atta through and holds back the lumps. Eratosthenes' sieve lets the primes through and catches the composites.",{"id":605,"type":606,"title":607,"items":608},"steps-sieve","steps","Sieving the numbers 1 to 50",[609,613,616,620,624,627,630],{"title":610,"tag":611,"text":612},"Write the grid","1–50","Write the numbers 1 to 50 in rows of 10.",{"title":614,"tag":403,"text":615},"Cross out 1","1 is neither prime nor composite, so cross it out.",{"title":617,"tag":618,"text":619},"Circle 2","first prime","Circle 2. Then cross out every other multiple of 2: 4, 6, 8, … up to 50. They are all composite.",{"title":621,"tag":622,"text":623},"Circle 3","next prime","The next number not crossed out is 3. Circle it and cross out its multiples: 6, 9, 12, … (some are already gone).",{"title":625,"tag":622,"text":626},"Circle 5","4 is already crossed out, so the next is 5. Circle it and cross out 10, 15, 20, … 50.",{"title":628,"tag":622,"text":629},"Circle 7","Circle 7 and cross out its multiples. The only new one is 49 (7 × 7). All the others were already crossed out.",{"title":631,"tag":632,"text":633},"Circle the rest","15 primes","Every number left standing is prime. Circle them all.",{"id":635,"type":102,"prompt":636,"options":637,"explanation":644},"pred-sieve-2","In the grid from 1 to 50, you cross out 1 and then every multiple of 2 except 2 itself. **How many numbers are still standing?**",[638,640,641,643],{"id":106,"label":639},"25",{"id":109,"label":220},{"id":112,"label":642},"26",{"id":183,"label":439},"**25.** The numbers 1 to 50 contain 25 even numbers and 25 odd numbers. You keep the even number 2 and cross out the other 24 evens. You cross out the odd number 1. So what remains is 2 plus the 24 odd numbers from 3 to 49: **25 numbers**. In one step, you have removed half the grid. The later primes have much less to do.",{"id":646,"type":194,"component":647,"componentVersion":5,"config":648,"objective":654,"textAlternative":655,"help":656},"lab-sieve-50","prime-sieve",{"max":649,"columns":650,"modes":651,"rounds":653},50,10,[652],"sieve",5,"Run the Sieve of Eratosthenes on 1 to 50 and watch the composites get crossed out, leaving only the primes.","The lab shows the numbers 1 to 50 in rows of 10. You tap a prime and every one of its multiples is crossed out.\n\n1. 1 is greyed out first: it is neither prime nor composite.\n2. Tap **2**. The even numbers 4, 6, 8, … 50 are crossed out: 24 numbers go.\n3. Tap **3**. Its multiples that are still standing go: 9, 15, 21, 27, 33, 39, 45.\n4. Tap **5**. New crossings: 25 and 35.\n5. Tap **7**. Only one new number is crossed out: 49.\n6. Now nothing more changes, even if you try 11 or 13, because their multiples up to 50 were already crossed out.\n\nThe 15 numbers left are the primes up to 50: **2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47**. On the grid, the even columns (except 2) and the column ending in 5 (except 5) are empty of primes.",{"simplerExplanation":657,"hints":658},"Each prime you tap knocks out its own times table. What is left over at the end can only be divided by 1 and itself, so it is prime.",[659,660],"Start with 2, then always tap the smallest number that is still standing.","After 7, does tapping 11 cross out anything new?",{"id":662,"type":47,"variant":48,"title":663,"markdown":664},"obs-sieve","Why we could stop at 7","When you tapped 7, only one new number, 49, got crossed out. After that, nothing new happened. The next prime is 11, and 11 × 11 = 121, which is bigger than 50. Any smaller multiple of 11, like 22 or 33, had already been caught by 2 or 3. You will find out exactly why this happens in the Understand layer.",{"id":666,"type":43,"markdown":667},"primes-100","If you sieve all the way to 100, you find **25 prime numbers**:\n\n**2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97**\n\nIt is worth learning the primes up to 30 by heart (there are ten of them), and knowing where to find the rest.",{"id":669,"type":194,"component":647,"componentVersion":5,"config":670,"objective":674,"textAlternative":675,"help":676},"lab-hunt-100",{"max":671,"columns":650,"modes":672,"rounds":92},100,[673],"hunt","Hunt for primes on a 1 to 100 grid: tap every prime in a row before the round ends.","In the hunt game, the lab shows the numbers 1 to 100 in rows of 10 and asks you to tap all the primes in one row (for example, 31 to 40) as quickly as you can. Each round scores points; tapping a composite costs a point.\n\nThe answers, row by row:\n- 1–10: 2, 3, 5, 7\n- 11–20: 11, 13, 17, 19\n- 21–30: 23, 29\n- 31–40: 31, 37\n- 41–50: 41, 43, 47\n- 51–60: 53, 59\n- 61–70: 61, 67\n- 71–80: 71, 73, 79\n- 81–90: 83, 89\n- 91–100: 97\n\nThe hardest traps are 51 (3 × 17), 57 (3 × 19), 87 (3 × 29) and 91 (7 × 13). Notice that the first two rows have four primes each but the last row has only one: primes thin out as numbers grow.",{"hints":677},[678,679],"Skip every even number except 2, and every number ending in 5 except 5.","Check the rest against 3 and 7.",{"id":681,"type":47,"variant":572,"title":682,"markdown":683},"ex-eratosthenes","The librarian who measured the Earth","Eratosthenes of Cyrene (about 276 to 194 BCE) ran the great Library of Alexandria. He is also famous for estimating the size of the whole Earth, using the lengths of shadows in two Egyptian cities on the same day and some clever geometry. His estimate was remarkably close to the modern value. A mathematician who could measure a planet with shadows also found the neatest way to catch primes.",{"id":685,"type":87,"itemId":686,"prompt":687,"check":688,"hints":689,"feedback":692},"prac-count-primes-30","prime-and-composite.discover-primes-to-30","How many prime numbers are there from **1 to 30**?",{"kind":91,"answer":650,"tolerance":93},[690,691],"List them: 2, 3, 5, 7, …","Remember 1 is not prime.",{"correct":693,"incorrect":694},"Yes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Ten primes.","There are 10: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. Did you include 1 (not prime) or 27 (= 3 × 9)?",{"id":696,"type":53,"title":697,"eyebrow":698,"navLabel":699},"ch8","Prime partners: twin primes and co-primes","Chapter 08","8 Twins and co-primes",{"id":701,"type":43,"markdown":702},"twin-prose","Look at the primes up to 100 again. Some of them sit very close together: **11 and 13**, **17 and 19**, **29 and 31**. Each pair differs by just 2. Primes like these are called **twin primes**.\n\nUp to 100 there are **8 pairs** of twin primes: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73).\n\n(Why not a gap of 1? Out of any two numbers next to each other, one is even, and the only even prime is 2. So the only primes that differ by 1 are 2 and 3.)\n\nNobody knows whether twin primes go on forever. Mathematicians have been trying to find out for well over a hundred years. It is one of the most famous **unsolved** questions in mathematics.",{"id":704,"type":87,"itemId":705,"prompt":706,"check":707,"hints":708,"feedback":711},"prac-twins-50","prime-and-composite.discover-twins-below-50","How many pairs of **twin primes** are there with both numbers below 50?",{"kind":91,"answer":92,"tolerance":93},[709,710],"Go through the primes up to 50 and look for neighbours that differ by 2.","Start with (3, 5) and (5, 7).",{"correct":712,"incorrect":713},"Yes: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31) and (41, 43). Six pairs.","There are six: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43). Note that 5 belongs to two pairs. (23, 25) is not a pair, because 25 = 5 × 5.",{"id":715,"type":43,"markdown":716},"coprime-prose","There is one more partner idea. Two numbers are **co-prime** when the **only factor they share is 1**.\n\n- **8 and 15**: factors of 8 are 1, 2, 4, 8. Factors of 15 are 1, 3, 5, 15. The only one in both lists is 1. So 8 and 15 are co-prime, even though **neither of them is prime**.\n- **8 and 12**: both lists contain 1, 2 and 4. They share more than 1, so they are **not** co-prime.\n\nCo-prime is about a **pair** of numbers getting along with no shared factors. Prime is about **one** number on its own.",{"id":718,"type":47,"variant":157,"title":719,"markdown":720},"misc-coprime","“Co-prime numbers must both be prime”","Not at all. 8 and 15, 4 and 9, and 10 and 21 are co-prime pairs made of composite numbers. What matters is that the pair shares no factor except 1. Two different primes, like 5 and 7, are always co-prime, but many other pairs are too.",{"id":722,"type":87,"itemId":723,"prompt":724,"check":725,"hints":736,"feedback":739},"prac-coprime","prime-and-composite.discover-coprime-pair","Which pair of numbers is **co-prime**?",{"kind":174,"options":726,"correct":735},[727,729,731,733],{"id":106,"label":728},"6 and 9",{"id":109,"label":730},"4 and 9",{"id":112,"label":732},"10 and 15",{"id":183,"label":734},"12 and 18",[109],[737,738],"List the factors of both numbers and look for a shared one bigger than 1.","Factors of 4: 1, 2, 4. Factors of 9: 1, 3, 9.",{"correct":740,"incorrect":741},"4 and 9 share only the factor 1, so they are co-prime, even though both are composite.","The co-prime pair is 4 and 9. 6 and 9 share 3; 10 and 15 share 5; 12 and 18 share 2, 3 and 6.",{"id":743,"type":53,"title":744,"eyebrow":745,"navLabel":746},"ch9","Primes around you","Chapter 09","9 Primes around you",{"id":748,"type":43,"markdown":749},"cicada-prose","Primes are not only found in maths books. In parts of the United States, some kinds of **periodical cicadas** (insects a little like large crickets) spend years underground and then come out all at once, in huge numbers. Some kinds appear every **13 years** and others every **17 years**. Both are prime numbers.\n\nScientists think this may help the cicadas avoid predators that have shorter life cycles. A predator that booms every 2, 3, 4 or 6 years will only rarely boom in the same year as a cicada with a prime cycle. You will do the arithmetic for this in the Extend layer.",{"id":751,"type":752,"title":753,"prompt":754,"options":755},"explorer-around","explorer","Where primes and factors turn up","Pick a place to see how factors or primes are involved.",[756,766,778,786],{"id":757,"label":758,"chain":759,"note":765},"assembly","School assembly",[760,761,762,763,764],"36 students","Factor pairs of 36","1 × 36, 2 × 18, 3 × 12","4 × 9, 6 × 6","Choose a neat block","A PE teacher arranging 36 students in equal rows can choose any factor pair of 36. 6 rows of 6 make a square. If one student is absent, 35 = 5 × 7 still works, but if 37 turn up there is no neat block at all, because 37 is prime.",{"id":767,"label":768,"chain":769,"badge":774,"note":777},"cicadas","Cicadas",[770,771,772,773],"Live underground","13 or 17 years","Prime cycles","Rarely meet predators",{"text":775,"tone":776},"Prime numbers in nature","yes","Periodical cicadas in North America emerge every 13 or 17 years. A cycle that has no small factors seldom lines up with the cycles of predators. The 13-year and 17-year broods themselves only come out in the same year once every 221 years, because 13 × 17 = 221.",{"id":779,"label":780,"chain":781,"note":785},"kirana","Kirana packing",[782,783,784],"24 soaps","Packs of 2, 3, 4, 6, 8 or 12","No loose soaps","A shopkeeper with 24 soaps can make equal packs of 1, 2, 3, 4, 6, 8, 12 or 24, because those are the factors of 24. With 23 soaps she can only sell them singly or as one big pack of 23.",{"id":787,"label":788,"chain":789,"badge":795,"note":797},"online","Online payments",[790,791,792,793,794],"Huge prime numbers","Multiply two of them","Product easy to make","Very hard to split","Keeps secrets safe",{"text":796,"tone":776},"Primes keep data safe","Some of the security systems behind online shopping and banking are built on a simple fact: multiplying two enormous primes is quick for a computer, but splitting the answer back into those two primes can take so long that nobody can do it. The Extend layer explains more.",{"id":799,"type":800,"conceptId":801,"relation":802,"explanation":803},"conn-four-ops","connection","four-operations","helps_understand","Finding factors is careful division: a factor is any number that divides with remainder 0.",{"id":805,"type":800,"conceptId":806,"relation":802,"explanation":807},"conn-hcf","hcf-and-lcm","Factors lead to common factors and the HCF; multiples lead to common multiples and the LCM.",{"id":809,"type":800,"conceptId":810,"relation":811,"explanation":812},"conn-shapes","shape-and-space","related_to","Factor pairs are the side lengths of every rectangle you can make from a number of square tiles.",{"id":814,"type":53,"title":815,"eyebrow":816,"navLabel":817},"ch10","Words and a quick check","Chapter 10","10 Words and quiz",{"id":819,"type":820,"title":821,"terms":822},"gloss-discover","glossary","Words to know",[823,827,831,835,839,843,847,851,854,858,862,865,869,873],{"term":824,"meaning":825,"example":826},"factor","A number that divides another number exactly, leaving no remainder.","The factors of 10 are 1, 2, 5 and 10.",{"term":828,"meaning":829,"example":830},"multiple","The result of multiplying a number by 1, 2, 3, 4 and so on. A number’s times table.","Multiples of 5: 5, 10, 15, 20, …",{"term":832,"meaning":833,"example":834},"divides exactly","Goes into a number with no remainder. Also said as \"is divisible by\".","15 is divisible by 3, because 15 ÷ 3 = 5.",{"term":836,"meaning":837,"example":838},"factor pair","Two numbers that multiply to give the number.","3 and 8 are a factor pair of 24.",{"term":840,"meaning":841,"example":842},"remainder","What is left over when a number cannot be shared equally.","17 ÷ 5 = 3 remainder 2.",{"term":844,"meaning":845,"example":846},"prime number","A counting number with exactly two different factors: 1 and itself.","2, 3, 5, 7, 11, 13",{"term":848,"meaning":849,"example":850},"composite number","A counting number with more than two factors.","4, 6, 8, 9, 10, 12",{"term":852,"meaning":853},"neither prime nor composite","The number 1, which has only one factor.",{"term":855,"meaning":856,"example":857},"twin primes","Two prime numbers that differ by 2.","11 and 13; 41 and 43",{"term":859,"meaning":860,"example":861},"co-prime numbers","Two numbers whose only common factor is 1. Also called relatively prime.","8 and 15",{"term":863,"meaning":864},"Sieve of Eratosthenes","A way to find all primes up to a number by crossing out the multiples of each prime in turn.",{"term":866,"meaning":867,"example":868},"even number","A whole number that is a multiple of 2.","2, 4, 6, 8",{"term":870,"meaning":871,"example":872},"odd number","A whole number that is not a multiple of 2.","1, 3, 5, 7",{"term":874,"meaning":875,"example":876},"array","Objects arranged in equal rows and columns.","3 rows of 4 = 12",{"id":878,"type":879,"title":880,"questions":881},"quiz-discover","quiz","Quick check: factors, multiples and primes",[882,893,904,917,926,934,947,960,971,983],{"itemId":883,"prompt":884,"options":885,"correct":109,"why":892},"prime-and-composite.discover-q-factor-of-15","Which of these is a **factor** of 15?",[886,887,888,890],{"id":106,"label":248},{"id":109,"label":413},{"id":112,"label":889},"30",{"id":183,"label":891},"45","15 ÷ 5 = 3 with no remainder. 30 and 45 are multiples of 15, not factors.",{"itemId":894,"prompt":895,"options":896,"correct":112,"why":903},"prime-and-composite.discover-q-multiple-of-7","Which of these is **not** a multiple of 7?",[897,898,899,901],{"id":106,"label":428},{"id":109,"label":458},{"id":112,"label":900},"47",{"id":183,"label":902},"63","47 ÷ 7 = 6 remainder 5. The others are 7 × 2, 7 × 5 and 7 × 9.",{"itemId":905,"prompt":906,"options":907,"correct":109,"why":916},"prime-and-composite.discover-q-prime-def","A prime number has…",[908,910,912,914],{"id":106,"label":909},"exactly one factor",{"id":109,"label":911},"exactly two factors",{"id":112,"label":913},"more than two factors",{"id":183,"label":915},"no factors","A prime has exactly two factors: 1 and itself.",{"itemId":918,"prompt":919,"options":920,"correct":112,"why":925},"prime-and-composite.discover-q-smallest-composite","What is the **smallest** composite number?",[921,922,923,924],{"id":106,"label":142},{"id":109,"label":248},{"id":112,"label":276},{"id":183,"label":204},"4 has factors 1, 2 and 4. 1 is neither, 2 is prime, and 3 is prime, so 4 is the first composite.",{"itemId":927,"prompt":928,"options":929,"correct":112,"why":933},"prime-and-composite.discover-q-one","The number 1 is…",[930,931,932],{"id":106,"label":406},{"id":109,"label":411},{"id":112,"label":852},"1 has only one factor, so it fits neither definition.",{"itemId":935,"prompt":936,"options":937,"correct":106,"why":946},"prime-and-composite.discover-q-rect-19","A gardener has **19** saplings. How many different rectangles (at least 2 rows of at least 2) can she plant?",[938,940,942,944],{"id":106,"label":939},"None",{"id":109,"label":941},"One",{"id":112,"label":943},"Two",{"id":183,"label":945},"Three","19 is prime, so the only arrangement is a single row of 19.",{"itemId":948,"prompt":949,"options":950,"correct":112,"why":959},"prime-and-composite.discover-q-twin","Which pair are **twin primes**?",[951,953,955,957],{"id":106,"label":952},"13 and 15",{"id":109,"label":954},"23 and 29",{"id":112,"label":956},"41 and 43",{"id":183,"label":958},"1 and 3","41 and 43 are both prime and differ by 2. 15 = 3 × 5, 23 and 29 differ by 6, and 1 is not prime.",{"itemId":961,"prompt":962,"options":963,"correct":106,"why":970},"prime-and-composite.discover-q-coprime","Are 9 and 16 co-prime?",[964,966,968],{"id":106,"label":965},"Yes: they share only the factor 1",{"id":109,"label":967},"No: neither is prime",{"id":112,"label":969},"No: both are square numbers","Factors of 9: 1, 3, 9. Factors of 16: 1, 2, 4, 8, 16. Only 1 is shared, so they are co-prime. They do not need to be prime themselves.",{"itemId":972,"prompt":973,"options":974,"correct":109,"why":982},"prime-and-composite.discover-q-even-prime","How many **even** prime numbers are there?",[975,976,978,980],{"id":106,"label":939},{"id":109,"label":977},"Exactly one",{"id":112,"label":979},"Exactly two",{"id":183,"label":981},"Infinitely many","Only 2. Every larger even number also has 2 as a factor.",{"itemId":984,"prompt":985,"options":986,"correct":109,"why":991},"prime-and-composite.discover-q-sieve","When you sieve 1 to 50, which is the **last** prime whose multiples cross out anything new?",[987,988,989,990],{"id":106,"label":413},{"id":109,"label":251},{"id":112,"label":262},{"id":183,"label":900},"7 crosses out 49. The multiples of 11 below 50 (22, 33, 44) were already crossed out by 2 and 3.",{"id":993,"type":994,"prompt":995},"reflect-discover","reflection","You have 60 sweets and want to share them into equal bags with none left over. List every bag size you could use. Then imagine you had 61 sweets instead. What changes, and why?",{"id":997,"type":998,"title":999,"points":1000},"cheat-discover","summary","Cheat sheet",[1001,1002,1003,1004,1005,1006,1007,1008,1009,1010,1011],"**Factor:** divides a number exactly. Factors of 12: 1, 2, 3, 4, 6, 12. A number has only a few factors.","**Multiple:** the number’s times table. Multiples of 12: 12, 24, 36, … They go on forever.","**3 is a factor of 12** means the same as **12 is a multiple of 3**.","**Factor pairs** are the sides of the rectangles you can make: 12 → 1 × 12, 2 × 6, 3 × 4.","**Prime:** exactly two factors (1 and itself). Only one rectangle: a single strip.","**Composite:** more than two factors. The smallest is 4.","**1** is neither prime nor composite. **2** is the only even prime.","**Sieve of Eratosthenes:** cross out the multiples of 2, 3, 5, 7, … and the primes are left.","**Primes up to 30:** 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. There are **25** primes up to 100.","**Twin primes** differ by 2: (3, 5), (5, 7), (11, 13), (17, 19), … Nobody knows if they go on forever.","**Co-prime:** a pair sharing only the factor 1, like 8 and 15. They need not be primes.",{"id":1013,"type":1014,"sourceIds":1015},"sources-discover","sources",[1016,1017,1018,1019,1020,1021],"prime-and-composite-ncert-class6-playing-numbers","prime-and-composite-ncert-class6-prime-time","prime-and-composite-khan-primes","prime-and-composite-mathisfun-prime","prime-and-composite-britannica-eratosthenes-sieve","prime-and-composite-wiki-periodical-cicada",[1016,1017,1018,1019,1020,1021],"needs_review",{"generatedBy":1025,"notes":1026},"claude-code","Draft generated locally with Python-checked numbers; pending owner review.","170098e08e2f33e258339a77b982c064e30a3508a85e0fff0d3d2559c1bddabe",{"logic:practice":1029,"component:match-pairs@1":1030,"component:sort-game@1":1031,"component:prime-sieve@1":1032,"source:prime-and-composite-britannica-eratosthenes-sieve":1033,"source:prime-and-composite-khan-primes":1034,"source:prime-and-composite-mathisfun-prime":1035,"source:prime-and-composite-ncert-class6-playing-numbers":1036,"source:prime-and-composite-ncert-class6-prime-time":1037,"source:prime-and-composite-wiki-periodical-cicada":1038},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","76baccc44f3b2ac333537045fea84801ea3c46f2d75f3db8a02355ef0f267041","af8e7bbdca2e2859b452689299eeef54805896728cef8a361ea1ec0dd63d12d4","7199c0a10c6364abf7af30d1869f37167826d837ce97a5b9cb0d8be7f2322212","ae2cb5b494e894b59fc2f34095d5ba7519d897f66195dc8bef83cbbcffa674b1","d766224fdc533b260af62a8660196dabe3a5e506c901ab99ee8eba647c9a7616","f6edb540745600a1873f0a5851335bf576bd243816f181e62085831fd55ac4ce","16058ea23b50d38df9f169941d978b866e6200e16f02efd314be2f5ffb56ea20",{"state":1040,"reviewer":1041,"selfReview":1042,"reviewedAt":1043,"method":1044},"approved","The library owner",true,"2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597856]