[{"data":1,"prerenderedAt":1138},["ShallowReactive",2],{"layer:prime-and-composite:understand":3},{"layer":4,"contentHash":1119,"dependencyHashes":1120,"approval":1132,"releaseId":1137},{"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":1114,"reviewStatus":1115,"authoring":1116},1,"prime-and-composite","en","understand","Factors, primes and how to test them","Precise definitions, reliable methods and the mix-ups to avoid","Find every factor with the factor-pair method, sieve to 100 and see why you can stop at 7, test any number for primality by trial division up to its square root, use divisibility rules, and meet twin primes, co-primes and factor trees.",[13,14,15,16,17],"Use the precise language of factors, multiples, divisors and divisibility.","List every factor of a number using factor pairs and explain when to stop.","Sieve to 100 and test a number for primality by dividing by primes up to its square root.","Apply divisibility rules for 2, 3, 4, 5, 6, 8, 9, 10 and 11, and combine them only with co-prime pieces.","Identify twin primes and co-prime pairs, and find prime factorisations with factor trees.",45,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Understand",{"label":26,"value":27},"Reading time","≈ 45 minutes",{"label":29,"value":30},"Prior knowledge","Times tables; Discover layer",{"label":32,"value":33},"Chapters","11",{"label":35,"value":36},"Labs","Sieve, digit-sum sort, co-prime sort, factor tree, match",{"label":38,"value":39},"Key rule","Try primes p while p × p ≤ n",[41,45,51,57,60,85,90,117,122,125,139,149,189,193,207,212,215,224,230,235,238,258,263,267,293,298,301,325,342,380,391,396,399,434,445,456,468,489,494,497,542,555,622,643,648,653,656,679,684,687,697,763,768,771,781,793,813,849,870,875,880,884,953,1084,1088,1104],{"id":42,"type":43,"markdown":44},"intro-understand","prose","In Discover you met factors, multiples, primes and composites through laddoos, tiles and a sieve. This layer makes those ideas **precise** and gives you **reliable methods**:\n\n- how to find **every** factor of a number without missing any, and know when to stop;\n- how to decide whether a large number like 221 or 437 is prime, with the least possible work;\n- the **divisibility rules** for 2, 3, 4, 5, 6, 8, 9, 10 and 11;\n- exactly what **twin primes** and **co-prime numbers** are, with their tricky cases;\n- how to break any composite number into primes with a **factor tree**.\n\nAlong the way you will meet the mix-ups that catch most learners, so you can avoid them.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"how-to-use-u","callout","observation","How to use this lesson","Each chapter has a method, at least one worked example and a quick practice. Try every practice before reading the feedback. Use the labs to check yourself: they generate new numbers every time.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch1","chapter","The exact language of factors","Chapter 01","1 Exact language",{"id":58,"type":43,"markdown":59},"lang-prose","Mathematicians use several phrases for the same fact. If **3 × 5 = 15**, then all of these are true:\n\n- **3 is a factor of 15** (also: 3 is a **divisor** of 15).\n- **15 is a multiple of 3.**\n- **15 is divisible by 3** (15 ÷ 3 = 5 with remainder 0).\n- **3 divides 15.** (Older books and later maths write this as 3 | 15, read \"3 divides 15\". The vertical bar is not a division sign.)\n\nAll four sentences are about **whole numbers** and **exact division**. We say 2 is **not** a factor of 15, even though 15 ÷ 2 = 7.5, because 7.5 is not a whole number: sharing 15 into 2 equal whole groups leaves 1 over.",{"id":61,"type":62,"tone":63,"items":64},"spec-lang","spec","blue",[65,69,73,77,81],{"label":66,"big":67,"value":68},"Factor \u002F divisor","a × b = n","a and b are factors of n. They divide n exactly.",{"label":70,"big":71,"value":72},"Multiple","n = a × k","n is a multiple of a when k is a counting number.",{"label":74,"big":75,"value":76},"Divisible","remainder 0","n is divisible by a when n ÷ a leaves no remainder.",{"label":78,"big":79,"value":80},"Factor pair","(a, b)","Two factors whose product is n, e.g. (4, 9) for 36.",{"label":82,"big":83,"value":84},"Proper factor","less than n","Any factor except the number itself. Proper factors of 12: 1, 2, 3, 4, 6.",{"id":86,"type":47,"variant":87,"title":88,"markdown":89},"nuance-zero","nuance","What about 0?","0 is a multiple of every number, because a × 0 = 0. But in school lists of multiples we start from the number itself (4, 8, 12, …), so 0 is left out.\n\nPrimes and composites are only defined for counting numbers **greater than 1**. 0 is neither prime nor composite, and so is 1. Also, you can never divide by 0, so 0 is never a factor of anything.",{"id":91,"type":92,"itemId":93,"prompt":94,"check":95,"hints":111,"feedback":114},"prac-lang","practice","prime-and-composite.understand-same-meaning","Which sentence means the **same** as \"28 is a multiple of 7\"?",{"kind":96,"options":97,"correct":110},"choice",[98,101,104,107],{"id":99,"label":100},"a","28 is a factor of 7",{"id":102,"label":103},"b","7 is divisible by 28",{"id":105,"label":106},"c","7 is a factor of 28",{"id":108,"label":109},"d","7 is a multiple of 28",[105],[112,113],"Write it as a multiplication: 7 × 4 = 28.","Which number is the smaller \"building block\"?",{"correct":115,"incorrect":116},"Right. 7 × 4 = 28, so 7 is a factor of 28 and 28 is a multiple of 7.","From 7 × 4 = 28: 7 is a factor of 28, 28 is divisible by 7, and 28 is a multiple of 7. The other sentences put the numbers the wrong way round.",{"id":118,"type":53,"title":119,"eyebrow":120,"navLabel":121},"ch2","Finding every factor, and knowing when to stop","Chapter 02","2 All the factors",{"id":123,"type":43,"markdown":124},"pairs-method","The safest way to list all factors is the **factor-pair method**:\n\n1. Write 1 and the number itself as the first pair.\n2. Try 2, 3, 4, 5, … in order. Whenever one divides exactly, write it with its partner (the answer of the division).\n3. **Stop** when the number you are trying is bigger than its partner would be, in other words, when the pairs meet in the middle.\n\nThe stopping rule is what makes this method fast. For 100 you only need to try up to 10, because 10 × 10 = 100. Any factor bigger than 10 must pair with a factor smaller than 10, and you have already found all of those.",{"id":126,"type":127,"title":128,"problem":129,"steps":130},"we-48","worked_example","All the factors of 48","List every factor of 48 using factor pairs.",[131,132,133,134,135,136,137,138],"1 × 48. Write **1, 48**.","48 is even: 2 × 24. Write **2, 24**.","Digit sum 4 + 8 = 12, a multiple of 3, so 3 divides 48: 3 × 16. Write **3, 16**.","4 × 12 = 48. Write **4, 12**.","5 does not divide 48 (it does not end in 0 or 5).","6 × 8 = 48. Write **6, 8**.","7: 7 × 6 = 42 and 7 × 7 = 49, so no. The next number to try, 7, is already bigger than its would-be partner (48 ÷ 7 is less than 7), so **stop**.","Factors of 48 in order: **1, 2, 3, 4, 6, 8, 12, 16, 24, 48**. That is 10 factors in 5 pairs.",{"id":140,"type":127,"title":141,"problem":142,"steps":143},"we-36","A square number has a lonely factor","List every factor of 36.",[144,145,146,147,148],"Pairs: 1 × 36, 2 × 18, 3 × 12, 4 × 9.","5 does not divide 36.","6 × 6 = 36. The pair is a number with itself, so we write 6 only once.","Factors of 36: **1, 2, 3, 4, 6, 9, 12, 18, 36**. That is 9 factors, an **odd** number.","Only **square numbers** (1, 4, 9, 16, 25, 36, …) have an odd number of factors, because only they have a factor that pairs with itself.",{"id":150,"type":151,"caption":152,"columns":153,"rows":157},"table-factor-counts","table","How many factors? Some factor-rich numbers up to 100",[154,155,156],"Number","Factor pairs","Number of factors",[158,162,166,170,174,177,180,183,186],[159,160,161],"12","1 × 12, 2 × 6, 3 × 4","6",[163,164,165],"24","1 × 24, 2 × 12, 3 × 8, 4 × 6","8",[167,168,169],"36","1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6","9",[171,172,173],"48","1 × 48, 2 × 24, 3 × 16, 4 × 12, 6 × 8","10",[175,176,159],"60","1 × 60, 2 × 30, 3 × 20, 4 × 15, 5 × 12, 6 × 10",[178,179,159],"72","1 × 72, 2 × 36, 3 × 24, 4 × 18, 6 × 12, 8 × 9",[181,182,159],"84","1 × 84, 2 × 42, 3 × 28, 4 × 21, 6 × 14, 7 × 12",[184,185,159],"90","1 × 90, 2 × 45, 3 × 30, 5 × 18, 6 × 15, 9 × 10",[187,188,159],"96","1 × 96, 2 × 48, 3 × 32, 4 × 24, 6 × 16, 8 × 12",{"id":190,"type":47,"variant":48,"title":191,"markdown":192},"obs-60","Why 60 minutes and 360 degrees?","60 has 12 factors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60), which is why an hour splits neatly into halves, thirds, quarters, fifths, sixths, tenths, twelfths and more. The ancient Babylonians counted in 60s partly for this reason. A full turn of **360°** has 24 factors, so a circle divides evenly in very many ways. Numbers up to 100 with the most factors (12 each) are 60, 72, 84, 90 and 96.",{"id":194,"type":92,"itemId":195,"prompt":196,"check":197,"hints":201,"feedback":204},"prac-72","prime-and-composite.understand-factors-of-72","How many factors does **72** have?",{"kind":198,"answer":199,"tolerance":200},"number",12,0,[202,203],"Pairs: 1 × 72, 2 × 36, 3 × 24, …","Keep going until the pairs meet. 8 × 9 = 72 is the last pair.",{"correct":205,"incorrect":206},"Yes: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72. The six pairs are 1 × 72, 2 × 36, 3 × 24, 4 × 18, 6 × 12, 8 × 9.","The pairs are 1 × 72, 2 × 36, 3 × 24, 4 × 18, 6 × 12, 8 × 9, giving 12 factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.",{"id":208,"type":53,"title":209,"eyebrow":210,"navLabel":211},"ch3","Multiples, common factors and common multiples","Chapter 03","3 Common ones",{"id":213,"type":43,"markdown":214},"common-prose","When two numbers are in play, we often want what they **share**.\n\n- A **common factor** of 12 and 18 divides both. Factors of 12: 1, 2, 3, 4, 6, 12. Factors of 18: 1, 2, 3, 6, 9, 18. Common factors: **1, 2, 3, 6**. The largest, 6, is the **highest common factor (HCF)**.\n- A **common multiple** of 4 and 6 is in both times tables. Multiples of 4: 4, 8, 12, 16, 20, 24, … Multiples of 6: 6, 12, 18, 24, 30, … Common multiples: **12, 24, 36, …** The smallest, 12, is the **lowest common multiple (LCM)**.\n\nTwo numbers always have a finite list of common factors (1 is always on it) and an endless list of common multiples (their product is always on it). The HCF and LCM get their own topic; here they matter because they explain **co-prime numbers**, later in this layer.",{"id":216,"type":127,"title":217,"problem":218,"steps":219},"we-common","Packing for a school trip","A teacher has 24 bananas and 36 oranges. She wants to make identical fruit bags using all the fruit, with no fruit left over. What numbers of bags are possible, and what is the most?",[220,221,222,223],"The number of bags must divide 24 exactly: 1, 2, 3, 4, 6, 8, 12, 24.","It must also divide 36 exactly: 1, 2, 3, 4, 6, 9, 12, 18, 36.","Common factors: **1, 2, 3, 4, 6, 12**. Any of these numbers of bags works.","The most is **12 bags**, each with 24 ÷ 12 = 2 bananas and 36 ÷ 12 = 3 oranges.",{"id":225,"type":226,"conceptId":227,"relation":228,"explanation":229},"conn-hcf-u","connection","hcf-and-lcm","helps_understand","Common factors and common multiples lead straight to the HCF and LCM, which have their own topic.",{"id":231,"type":53,"title":232,"eyebrow":233,"navLabel":234},"ch4","Prime, composite or neither: the precise rules","Chapter 04","4 Precise rules",{"id":236,"type":43,"markdown":237},"pcn-prose","Now the three definitions can be stated exactly, for every counting number n:\n\n- **n is prime** if it has **exactly two** factors, 1 and n. That forces n to be bigger than 1.\n- **n is composite** if it has **more than two** factors. Equivalently, n can be written as a × b where both a and b are whole numbers bigger than 1.\n- **n = 1** has exactly **one** factor and is **neither**.\n\nA useful way to say the composite rule: a composite number has a factor that is **neither 1 nor itself**. Find one such factor and you have proved the number is composite. To prove a number is prime, you must show that **no** such factor exists, which is harder. The rest of this layer is about doing that efficiently.",{"id":239,"type":62,"tone":240,"items":241},"spec-pcn","amber",[242,246,250,254],{"label":243,"big":244,"value":245},"Neither","1","Exactly one factor. Not prime, not composite.",{"label":247,"big":248,"value":249},"Prime","2 factors","2, 3, 5, 7, 11, 13, … The smallest is 2, the only even one.",{"label":251,"big":252,"value":253},"Composite","3 or more","4, 6, 8, 9, 10, 12, … The smallest is 4.",{"label":255,"big":256,"value":257},"Every n ≥ 2","one or other","Every counting number from 2 upwards is either prime or composite, never both.",{"id":259,"type":47,"variant":260,"title":261,"markdown":262},"why-one","aha","Why mathematicians agreed 1 is not prime","It is a choice, but a very good one. The most important fact about primes is that every number bigger than 1 can be built by multiplying primes in **exactly one way** (apart from order): 60 = 2 × 2 × 3 × 5 and no other set of primes gives 60.\n\nIf 1 were a prime, that would break at once: 60 = 2 × 2 × 3 × 5 × 1 = 2 × 2 × 3 × 5 × 1 × 1, and so on forever. Leaving 1 out keeps the rule clean. Some older books did call 1 prime; modern mathematics does not.",{"id":264,"type":47,"variant":260,"title":265,"markdown":266},"why-two","Why 2 is the only even prime","Every even number is 2 × something. For 2 itself the \"something\" is 1, so its factors are only 1 and 2. For any bigger even number, 2 is a factor that is neither 1 nor the number, so it is composite. That is why every prime after 2 is odd, and why the sieve wipes out half the grid in its first step.",{"id":268,"type":92,"itemId":269,"prompt":270,"check":271,"hints":287,"feedback":290},"prac-pcn","prime-and-composite.understand-select-composites","Select **all** the composite numbers.",{"kind":96,"options":272,"correct":286},[273,275,277,279,281,284],{"id":99,"label":274},"27",{"id":102,"label":276},"29",{"id":105,"label":278},"51",{"id":108,"label":280},"53",{"id":282,"label":283},"e","57",{"id":285,"label":244},"f",[99,105,282],[288,289],"Look for a factor that is neither 1 nor the number.","Try dividing each by 3.",{"correct":291,"incorrect":292},"27 = 3 × 9, 51 = 3 × 17 and 57 = 3 × 19 are composite. 29 and 53 are prime; 1 is neither.","The composites are 27 (3 × 9), 51 (3 × 17) and 57 (3 × 19). 29 and 53 are prime and 1 is neither prime nor composite.",{"id":294,"type":53,"title":295,"eyebrow":296,"navLabel":297},"ch5","The Sieve of Eratosthenes to 100","Chapter 05","5 Sieve to 100",{"id":299,"type":43,"markdown":300},"sieve-100","To find every prime up to 100, write the numbers 1 to 100, cross out 1, then for each prime in turn cross out its multiples. The surprise is how early you can stop: **after 7**.\n\nHere is why. Every composite number up to 100 is a × b with a ≤ b. If both a and b were bigger than 10, then a × b would be bigger than 10 × 10 = 100. So every composite up to 100 has a factor that is **10 or less**, and therefore a **prime factor** of 10 or less: 2, 3, 5 or 7. The sieve with 2, 3, 5 and 7 catches all of them.\n\nThe same reasoning tells you where to **start** crossing out for each prime. For 7, start at 7 × 7 = 49: the smaller multiples 14, 21, 28, 35 and 42 have a smaller prime factor (2, 3 or 5) and are already crossed out.\n\nWhat remains are the **25 primes** up to 100: 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.",{"id":302,"type":151,"caption":303,"columns":304,"rows":308},"table-sieve-rounds","What each prime crosses out on the 1 to 100 grid (numbers not already crossed out)",[247,305,306,307],"Start at","New numbers crossed out","How many new",[309,314,318,322],[310,311,312,313],"2","4","4, 6, 8, … 100 (all even numbers after 2)","49",[315,169,316,317],"3","9, 15, 21, 27, 33, 39, 45, 51, 57, 63, 69, 75, 81, 87, 93, 99","16",[319,320,321,161],"5","25","25, 35, 55, 65, 85, 95",[323,313,324,315],"7","49, 77, 91",{"id":326,"type":327,"component":328,"componentVersion":5,"config":329,"objective":336,"textAlternative":337,"help":338},"lab-sieve-100","interactive","prime-sieve",{"max":330,"columns":331,"modes":332,"rounds":335},100,10,[333,334],"sieve","hunt",8,"Sieve 1 to 100, count how many new numbers each prime removes, then hunt for primes on the finished grid.","The lab shows the numbers 1 to 100 in rows of 10. In **sieve** mode you choose each prime in turn and its multiples are crossed out, with a counter for how many new numbers each prime removed.\n\n- 1 is crossed out first (neither prime nor composite).\n- 2 removes 49 numbers (4, 6, … 100).\n- 3 removes 16 more, starting at 9.\n- 5 removes 6 more: 25, 35, 55, 65, 85, 95.\n- 7 removes only 3 more: 49, 77, 91.\n- 11 removes nothing new, because 11 × 11 = 121 is past 100.\n\nThat leaves 100 − 1 − 49 − 16 − 6 − 3 = **25 primes**. In **hunt** mode you race to tap all the primes in a row. Useful patterns: after the first row, primes only appear in the columns ending in 1, 3, 7 and 9.",{"hints":339},[340,341],"Always pick the smallest number still standing: it is the next prime.","Watch the counter shrink: 49, 16, 6, 3, then 0.",{"id":343,"type":151,"caption":344,"columns":345,"rows":349},"table-decades","How many primes in each block of ten up to 100",[346,347,348],"Block","Primes","Count",[350,353,356,359,362,365,368,371,374,377],[351,352,311],"1–10","2, 3, 5, 7",[354,355,311],"11–20","11, 13, 17, 19",[357,358,310],"21–30","23, 29",[360,361,310],"31–40","31, 37",[363,364,315],"41–50","41, 43, 47",[366,367,310],"51–60","53, 59",[369,370,310],"61–70","61, 67",[372,373,315],"71–80","71, 73, 79",[375,376,310],"81–90","83, 89",[378,379,244],"91–100","97",{"id":381,"type":92,"itemId":382,"prompt":383,"check":384,"hints":385,"feedback":388},"prac-primes-50-100","prime-and-composite.understand-primes-51-to-100","How many primes are there **from 51 to 100**?",{"kind":198,"answer":331,"tolerance":200},[386,387],"Use the table of blocks of ten.","Add the counts for 51–60, 61–70, 71–80, 81–90 and 91–100.",{"correct":389,"incorrect":390},"Yes: 53, 59, 61, 67, 71, 73, 79, 83, 89, 97. Ten primes, compared with fifteen from 1 to 50.","Count 2 + 2 + 3 + 2 + 1 = 10: 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.",{"id":392,"type":53,"title":393,"eyebrow":394,"navLabel":395},"ch6","Is it prime? Trial division up to the square root","Chapter 06","6 Testing primes",{"id":397,"type":43,"markdown":398},"trial-prose","To test a single number n for being prime, you do not need to sieve. Use **trial division**:\n\n1. Divide n by the primes **2, 3, 5, 7, 11, 13, …** in order.\n2. If one of them divides exactly, n is **composite**. Stop.\n3. Stop trying once the prime **squared** is bigger than n. If nothing has divided n by then, n is **prime**.\n\nThe stopping rule is the sieve's reason again. If n = a × b with a ≤ b, then a × a ≤ a × b = n, so the smaller factor a is at most √n (the **square root** of n, the number that multiplied by itself gives n). If no prime up to √n divides n, then n has no factor pair at all except 1 × n.\n\nYou never need to try composite numbers like 4, 6 or 9. If 9 divided n, then 3 would too, and you would already have found it.",{"id":400,"type":151,"caption":401,"columns":402,"rows":406},"table-stop","Which primes do you need to try? Stop once the prime squared is bigger than the number",[403,404,405],"If the number is below…","Try these primes","Because",[407,409,412,415,418,422,426,430],[169,310,408],"3 × 3 = 9",[320,410,411],"2, 3","5 × 5 = 25",[313,413,414],"2, 3, 5","7 × 7 = 49",[416,352,417],"121","11 × 11 = 121",[419,420,421],"169","2, 3, 5, 7, 11","13 × 13 = 169",[423,424,425],"289","2, 3, 5, 7, 11, 13","17 × 17 = 289",[427,428,429],"361","primes up to 17","19 × 19 = 361",[431,432,433],"529","primes up to 19","23 × 23 = 529",{"id":435,"type":127,"title":436,"problem":437,"steps":438},"we-91","Is 91 prime?","Decide whether 91 is prime.",[439,440,441,442,443,444],"√91 is a little more than 9 (9 × 9 = 81, 10 × 10 = 100). So try the primes 2, 3, 5 and 7.","2: 91 is odd. No.","3: digit sum 9 + 1 = 10, not a multiple of 3. No.","5: does not end in 0 or 5. No.","7: 7 × 13 = 91. **Yes.**","**91 is composite**: 91 = 7 × 13. It is the most famous \"looks prime but isn’t\" number under 100.",{"id":446,"type":127,"title":447,"problem":448,"steps":449},"we-211","Is 211 prime?","Decide whether 211 is prime.",[450,451,452,453,454,455],"14 × 14 = 196 and 15 × 15 = 225, so √211 is between 14 and 15. Try primes up to 13: 2, 3, 5, 7, 11, 13.","2: odd. 3: digit sum 2 + 1 + 1 = 4. 5: ends in 1. None divide.","7: 7 × 30 = 210, so 211 = 7 × 30 + 1. Remainder 1. No.","11: alternating digit sum 1 − 1 + 2 = 2, not 0 or a multiple of 11. No.","13: 13 × 16 = 208, remainder 3. No.","No prime up to √211 divides it, so **211 is prime**. Six small checks were enough to settle a three-digit number.",{"id":457,"type":458,"prompt":459,"options":460,"explanation":467},"pred-221","prediction","Priya tests **221** by dividing by 2, 3, 5, 7 and 11. None of them works, so she says 221 is prime. Is she right?",[461,463,465],{"id":99,"label":462},"Yes: she tried enough primes",{"id":102,"label":464},"No: she stopped too early",{"id":105,"label":466},"No: she should also have tried 4, 6, 8 and 9","**No: she stopped too early.** √221 is between 14 and 15 (14 × 14 = 196, 15 × 15 = 225), so she must also try **13**. And 13 × 17 = 221, so 221 is composite.\n\nThe rule is to keep going until the prime squared passes the number: 11 × 11 = 121 is still less than 221, so 11 is not the last one to check; 13 × 13 = 169 is also less than 221; only 17 × 17 = 289 is past it. Trying 4, 6, 8 or 9 is never needed: if they divided 221, then 2 or 3 would have too.",{"id":469,"type":92,"itemId":470,"prompt":471,"check":472,"hints":483,"feedback":486},"prac-119","prime-and-composite.understand-is-119-prime","Is **119** prime?",{"kind":96,"options":473,"correct":482},[474,476,478,480],{"id":99,"label":475},"Yes",{"id":102,"label":477},"No: it is divisible by 7",{"id":105,"label":479},"No: it is divisible by 3",{"id":108,"label":481},"No: it is divisible by 11",[102],[484,485],"√119 is just under 11, so try 2, 3, 5 and 7.","7 × 17 = ?",{"correct":487,"incorrect":488},"Correct: 119 = 7 × 17.","119 = 7 × 17. It is odd, its digit sum 11 is not a multiple of 3, and it does not end in 0 or 5, but 7 divides it.",{"id":490,"type":53,"title":491,"eyebrow":492,"navLabel":493},"ch7","Divisibility rules: shortcuts for small factors","Chapter 07","7 Divisibility rules",{"id":495,"type":43,"markdown":496},"div-prose","Trial division is faster with **divisibility rules**: quick tests that tell you whether a small number divides a big one, often just by looking at its digits. In this layer you learn **how** to use them. The Deepen layer shows **why** each one works, using place value.",{"id":498,"type":151,"caption":499,"columns":500,"rows":505},"table-div-rules","Divisibility rules for 2, 3, 4, 5, 6, 8, 9, 10 and 11",[501,502,503,504],"Divisible by","Test","Example that passes","Example that fails",[506,510,514,518,522,526,530,534,538],[310,507,508,509],"Last digit is 0, 2, 4, 6 or 8","3,456","3,457",[315,511,512,513],"Sum of digits is divisible by 3","5,142 (5 + 1 + 4 + 2 = 12)","5,143 (sum 13)",[311,515,516,517],"Last two digits form a number divisible by 4","7,316 (16 = 4 × 4)","7,318 (18)",[319,519,520,521],"Last digit is 0 or 5","9,875","9,871",[161,523,524,525],"Divisible by 2 **and** by 3","4,524 (even; sum 15)","4,526 (sum 17)",[165,527,528,529],"Last three digits form a number divisible by 8","15,128 (128 = 8 × 16)","15,132 (132)",[169,531,532,533],"Sum of digits is divisible by 9","6,354 (sum 18)","6,355 (sum 19)",[173,535,536,537],"Last digit is 0","4,730","4,735",[33,539,540,541],"Alternating sum of digits (from the right, +, −, +, …) is 0 or a multiple of 11","9,284 (4 − 8 + 2 − 9 = −11)","9,285 (−10)",{"id":543,"type":127,"title":544,"problem":545,"steps":546},"we-div-7128","Testing 7,128 with every rule","Which of 2, 3, 4, 5, 6, 8, 9, 10 and 11 divide 7,128?",[547,548,549,550,551,552,553,554],"**2:** last digit 8 is even. Yes.","**3 and 9:** digit sum 7 + 1 + 2 + 8 = 18. 18 is divisible by 3 and by 9. Yes to both.","**4:** last two digits 28 = 4 × 7. Yes.","**5 and 10:** last digit is 8, not 0 or 5. No to both.","**6:** divisible by 2 and by 3. Yes.","**8:** last three digits 128 = 8 × 16. Yes.","**11:** from the right, 8 − 2 + 1 − 7 = 0. Yes!","So 7,128 is divisible by **2, 3, 4, 6, 8, 9 and 11**. Check: 7,128 = 8 × 891 = 8 × 81 × 11, and in primes 7,128 = 2³ × 3⁴ × 11.",{"id":556,"type":327,"component":557,"componentVersion":5,"config":558,"objective":616,"textAlternative":617,"help":618},"lab-div-sort","sort-game",{"prompt":559,"bins":560,"items":570,"seconds":200},"Use the digit-sum rule: is each number divisible by 9, divisible by 3 but not 9, or not divisible by 3?",[561,564,567],{"id":562,"label":563},"nine","Divisible by 9",{"id":565,"label":566},"three","By 3, not by 9",{"id":568,"label":569},"none","Not divisible by 3",[571,575,579,583,586,589,593,597,601,605,608,612],{"id":572,"label":573,"bin":562,"why":574},"d144","144","Digit sum 9 is a multiple of 9.",{"id":576,"label":577,"bin":565,"why":578},"d231","231","Digit sum 6 is a multiple of 3 but not of 9.",{"id":580,"label":581,"bin":568,"why":582},"d407","407","Digit sum 11 is not a multiple of 3.",{"id":584,"label":585,"bin":562,"why":574},"d1116","1,116",{"id":587,"label":588,"bin":565,"why":578},"d2022","2,022",{"id":590,"label":591,"bin":562,"why":592},"d3591","3,591","Digit sum 18 is a multiple of 9.",{"id":594,"label":595,"bin":568,"why":596},"d4444","4,444","Digit sum 16 is not a multiple of 3.",{"id":598,"label":599,"bin":568,"why":600},"d5000","5,000","Digit sum 5 is not a multiple of 3.",{"id":602,"label":603,"bin":565,"why":604},"d6789","6,789","Digit sum 30 is a multiple of 3 but not of 9.",{"id":606,"label":607,"bin":562,"why":592},"d8181","8,181",{"id":609,"label":610,"bin":568,"why":611},"d10010","10,010","Digit sum 2 is not a multiple of 3.",{"id":613,"label":614,"bin":565,"why":615},"d12345","12,345","Digit sum 15 is a multiple of 3 but not of 9.","Apply the digit-sum tests for 3 and 9 to sort numbers into three groups.","This game gives 12 numbers and three bins: **Divisible by 9**, **By 3 but not by 9**, and **Not divisible by 3**. Add the digits of each number and check the sum.\n\n- Divisible by 9: 144 (sum 9), 1,116 (sum 9), 3,591 (sum 18), 8,181 (sum 18).\n- By 3 but not 9: 231 (sum 6), 2,022 (sum 6), 6,789 (sum 30), 12,345 (sum 15).\n- Not divisible by 3: 407 (sum 11), 4,444 (sum 16), 5,000 (sum 5), 10,010 (sum 2).\n\nEvery multiple of 9 is also a multiple of 3, so a number whose digit sum is 9, 18 or 27 goes in the \"divisible by 9\" bin. A digit sum like 30 is a multiple of 3 but not of 9.",{"hints":619},[620,621],"Add the digits. If the sum is still big, add its digits again.","A digit sum of 9, 18, 27 means divisible by 9.",{"id":623,"type":92,"itemId":624,"prompt":625,"check":626,"hints":637,"feedback":640},"prac-div","prime-and-composite.understand-div-by-8","Which number is divisible by **8**?",{"kind":96,"options":627,"correct":636},[628,630,632,634],{"id":99,"label":629},"3,412",{"id":102,"label":631},"5,136",{"id":105,"label":633},"7,284",{"id":108,"label":635},"9,020",[102],[638,639],"Look only at the last three digits.","136 ÷ 8 = ?",{"correct":641,"incorrect":642},"136 = 8 × 17, so 5,136 is divisible by 8.","Check the last three digits: 412 ÷ 8 leaves 4, 136 = 8 × 17 exactly, 284 ÷ 8 leaves 4 and 20 ÷ 8 leaves 4. Only 5,136 works.",{"id":644,"type":47,"variant":645,"title":646,"markdown":647},"careful-6","careful","Combining rules only works with co-prime pieces","\"Divisible by 2 and by 3 means divisible by 6\" works because 2 and 3 share no factor except 1. But \"divisible by 2 and by 4 means divisible by 8\" is **false**: 12 is divisible by 2 and by 4 but not by 8. Similarly 18 is divisible by 3 and 6 but not by 18. For 12 use 3 and 4; for 18 use 2 and 9; for 24 use 3 and 8. In each case the two pieces are **co-prime**.",{"id":649,"type":53,"title":650,"eyebrow":651,"navLabel":652},"ch8","Twin primes and the only prime triplet","Chapter 08","8 Twin primes",{"id":654,"type":43,"markdown":655},"twin-u","**Twin primes** are two primes whose difference is exactly 2. Up to 100 there are 8 pairs:\n\n(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73).\n\nSome facts to be precise about:\n\n- (2, 3) is **not** a twin pair: they differ by 1. It is the only pair of primes that are next-door neighbours.\n- 5 belongs to two pairs, (3, 5) and (5, 7). That makes **3, 5, 7** the only set of three primes each 2 apart, called a **prime triplet** in this sense. The Investigate layer finds out why there can never be another one.\n- The number **between** a twin pair (4, 6, 12, 18, 30, 42, 60, 72) is always a multiple of 6 after the first pair. Check it for yourself.\n- Up to 1,000 there are 35 twin pairs. Nobody knows whether there are infinitely many: this is the **twin prime conjecture**, still unsolved.",{"id":657,"type":92,"itemId":658,"prompt":659,"check":660,"hints":673,"feedback":676},"prac-twin","prime-and-composite.understand-twin-pairs","Select **all** the twin prime pairs.",{"kind":96,"options":661,"correct":672},[662,664,666,668,670],{"id":99,"label":663},"(59, 61)",{"id":102,"label":665},"(61, 63)",{"id":105,"label":667},"(71, 73)",{"id":108,"label":669},"(87, 89)",{"id":282,"label":671},"(101, 103)",[99,105,282],[674,675],"Both numbers must be prime.","63 = 7 × 9 and 87 = 3 × 29.",{"correct":677,"incorrect":678},"(59, 61), (71, 73) and (101, 103) are twin primes.","The twin pairs are (59, 61), (71, 73) and (101, 103). 63 = 3 × 21 and 87 = 3 × 29 are composite, so (61, 63) and (87, 89) are not twins.",{"id":680,"type":53,"title":681,"eyebrow":682,"navLabel":683},"ch9","Co-prime numbers","Chapter 09","9 Co-primes",{"id":685,"type":43,"markdown":686},"coprime-u","Two numbers are **co-prime** (also called **relatively prime**) when their **highest common factor is 1**: they share no factor except 1.\n\nPrecise facts, each worth remembering:\n\n- Co-prime is a property of a **pair** (or a set) of numbers, never of one number.\n- The numbers **need not be prime**: 8 and 15 are co-prime, and so are 4 and 9, 14 and 25, 16 and 27.\n- **Two different primes** are always co-prime, since each has only itself and 1 as factors.\n- **Consecutive numbers**, like 20 and 21 or 99 and 100, are always co-prime. (A number that divided both would have to divide their difference, 1.)\n- **Two even numbers** are never co-prime: they share the factor 2.\n- **1 is co-prime to every number.**\n- A prime p and any number that is not a multiple of p are co-prime: 7 and 30 are, but 7 and 35 are not.",{"id":688,"type":127,"title":689,"problem":690,"steps":691},"we-coprime","Are 21 and 40 co-prime? Are 21 and 45?","Decide whether each pair is co-prime.",[692,693,694,695,696],"Factors of 21: 1, 3, 7, 21. Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40.","The only common factor is 1, so **21 and 40 are co-prime**, even though both are composite.","Factors of 45: 1, 3, 5, 9, 15, 45. Common factors of 21 and 45: 1 and 3.","They share 3, so **21 and 45 are not co-prime**.","Faster: 21 = 3 × 7 and 40 = 2³ × 5 share no prime, so they are co-prime. 45 = 3² × 5 shares the prime 3 with 21.",{"id":698,"type":327,"component":557,"componentVersion":5,"config":699,"objective":757,"textAlternative":758,"help":759},"lab-coprime-sort",{"prompt":700,"bins":701,"items":708,"seconds":200},"Co-prime or not? Decide whether each pair shares a factor bigger than 1.",[702,705],{"id":703,"label":704},"yes","Co-prime",{"id":706,"label":707},"no","Not co-prime",[709,713,717,721,725,729,733,737,741,745,749,753],{"id":710,"label":711,"bin":703,"why":712},"p8-15","8 and 15","HCF is 1. Neither is prime, but they share no factor.",{"id":714,"label":715,"bin":706,"why":716},"p6-9","6 and 9","Both are divisible by 3. Both are in the 3 times table.",{"id":718,"label":719,"bin":706,"why":720},"p13-26","13 and 26","Both are divisible by 13. 26 is a multiple of 13.",{"id":722,"label":723,"bin":703,"why":724},"p20-21","20 and 21","HCF is 1. Consecutive numbers are always co-prime.",{"id":726,"label":727,"bin":703,"why":728},"p1-50","1 and 50","HCF is 1. 1 is co-prime to every number.",{"id":730,"label":731,"bin":703,"why":732},"p12-35","12 and 35","HCF is 1. 12 = 2² × 3 and 35 = 5 × 7 share no prime.",{"id":734,"label":735,"bin":706,"why":736},"p14-21","14 and 21","Both are divisible by 7. They share the factor 7.",{"id":738,"label":739,"bin":703,"why":740},"p17-31","17 and 31","HCF is 1. Two different primes are always co-prime.",{"id":742,"label":743,"bin":706,"why":744},"p24-36","24 and 36","Both are divisible by 12. They share 12.",{"id":746,"label":747,"bin":703,"why":748},"p9-28","9 and 28","HCF is 1. 9 = 3² and 28 = 2² × 7.",{"id":750,"label":751,"bin":706,"why":752},"p16-30","16 and 30","Both are divisible by 2. Two even numbers always share 2.",{"id":754,"label":755,"bin":703,"why":756},"p25-49","25 and 49","HCF is 1. Squares of different primes: 5² and 7².","Decide whether pairs of numbers are co-prime by looking for a shared factor bigger than 1.","This game shows 12 pairs of numbers and two bins, **Co-prime** and **Not co-prime**.\n\n- **Co-prime** (HCF 1): 8 and 15; 20 and 21 (consecutive); 1 and 50; 12 and 35; 17 and 31 (two primes); 9 and 28; 25 and 49.\n- **Not co-prime:** 6 and 9 (share 3); 13 and 26 (share 13); 14 and 21 (share 7); 24 and 36 (share 12); 16 and 30 (share 2).\n\nMost of the co-prime pairs contain composite numbers. What matters is only whether the pair shares a factor.",{"hints":760},[761,762],"If both are even, they share 2: not co-prime.","Split each number into primes and look for a prime in both.",{"id":764,"type":53,"title":765,"eyebrow":766,"navLabel":767},"ch10","Breaking numbers into primes: factor trees","Chapter 10","10 Factor trees",{"id":769,"type":43,"markdown":770},"pf-prose","Every composite number can be split into smaller factors, and those can be split again, until only primes are left. Writing a number as a product of primes is called its **prime factorisation**.\n\nA **factor tree** is a neat way to do it. Write the number at the top. Split it into any factor pair (not 1 × itself). Keep splitting every composite branch. Stop when every leaf at the bottom is a prime. Then multiply the leaves together.\n\nYou can also use **repeated division**: divide by the smallest prime that works, write down the result, and repeat until you reach 1. This is sometimes called the **division ladder**.\n\nWe use **index form** to write repeated primes neatly: 2 × 2 × 2 is written 2³ (read \"2 cubed\" or \"2 to the power 3\").",{"id":772,"type":127,"title":773,"problem":774,"steps":775},"we-tree-60","Two different trees for 60","Find the prime factorisation of 60 using two different first splits.",[776,777,778,779,780],"Tree A: 60 → 6 × 10. Then 6 → 2 × 3 and 10 → 2 × 5. Leaves: 2, 3, 2, 5.","Tree B: 60 → 4 × 15. Then 4 → 2 × 2 and 15 → 3 × 5. Leaves: 2, 2, 3, 5.","Both trees end with the **same primes**: two 2s, one 3, one 5.","60 = 2 × 2 × 3 × 5 = **2² × 3 × 5**.","This is always true: however you start, you reach the same primes. The Deepen layer explains why this matters so much.",{"id":782,"type":127,"title":783,"problem":784,"steps":785},"we-ladder-360","The division ladder for 360","Find the prime factorisation of 360 by repeated division.",[786,787,788,789,790,791,792],"360 ÷ 2 = 180.","180 ÷ 2 = 90.","90 ÷ 2 = 45 (45 is odd, so move on from 2).","45 ÷ 3 = 15.","15 ÷ 3 = 5.","5 ÷ 5 = 1. Stop.","The divisors used were 2, 2, 2, 3, 3, 5. So 360 = **2³ × 3² × 5**.",{"id":794,"type":327,"component":795,"componentVersion":5,"config":796,"objective":807,"textAlternative":808,"help":809},"lab-tree-u","factor-tree",{"numbers":797,"showIndexForm":806},[199,798,799,800,801,802,803,804,805,330],18,30,36,48,60,72,84,90,true,"Build factor trees by splitting each number into factor pairs until every leaf is prime, then write the answer in index form.","This lab gives you a number at the top of a tree. You choose a factor pair to split it, then keep splitting any branch that is not prime. The lab rings each prime leaf and, at the end, writes the product in index form.\n\nThe numbers and their prime factorisations:\n- 12 = 2 × 2 × 3 = **2² × 3**\n- 18 = 2 × 3 × 3 = **2 × 3²**\n- 30 = 2 × 3 × 5 = **2 × 3 × 5**\n- 36 = 2 × 2 × 3 × 3 = **2² × 3²**\n- 48 = 2 × 2 × 2 × 2 × 3 = **2⁴ × 3**\n- 60 = 2 × 2 × 3 × 5 = **2² × 3 × 5**\n- 72 = 2 × 2 × 2 × 3 × 3 = **2³ × 3²**\n- 84 = 2 × 2 × 3 × 7 = **2² × 3 × 7**\n- 90 = 2 × 3 × 3 × 5 = **2 × 3² × 5**\n- 100 = 2 × 2 × 5 × 5 = **2² × 5²**\n\nTry starting 36 with 4 × 9 and again with 6 × 6 or 2 × 18: the leaves always come out as two 2s and two 3s.",{"hints":810},[811,812],"Any factor pair except 1 × the number is a legal split.","If you are stuck, split off a 2 or a 3.",{"id":814,"type":327,"component":815,"componentVersion":5,"config":816,"objective":843,"textAlternative":844,"help":845},"lab-match-pf","match-pairs",{"prompt":817,"mode":818,"pairs":819},"Match each number with its prime factorisation.","connect",[820,822,825,828,831,834,837,840],{"a":163,"b":821},"2³ × 3",{"a":823,"b":824},"28","2² × 7",{"a":826,"b":827},"45","3² × 5",{"a":829,"b":830},"50","2 × 5²",{"a":832,"b":833},"63","3² × 7",{"a":835,"b":836},"64","2⁶",{"a":838,"b":839},"75","3 × 5²",{"a":841,"b":842},"98","2 × 7²","Connect each number to its prime factorisation written in index form.","This connect game has 8 numbers on one side and 8 factorisations on the other. Draw a line between each matching pair.\n\n- 24 → 2³ × 3\n- 28 → 2² × 7\n- 45 → 3² × 5\n- 50 → 2 × 5²\n- 63 → 3² × 7\n- 64 → 2⁶\n- 75 → 3 × 5²\n- 98 → 2 × 7²\n\nA quick check is to multiply the primes back together: for 98, 2 × 7² = 2 × 49 = 98. Watch out for 64, which is 2 multiplied by itself six times: 2⁶.",{"hints":846},[847,848],"Even numbers must include a 2.","Multiply the primes back to check.",{"id":850,"type":92,"itemId":851,"prompt":852,"check":853,"hints":864,"feedback":867},"prac-pf","prime-and-composite.understand-pf-84","What is the prime factorisation of **84**?",{"kind":96,"options":854,"correct":863},[855,857,859,861],{"id":99,"label":856},"2 × 42",{"id":102,"label":858},"4 × 3 × 7",{"id":105,"label":860},"2² × 3 × 7",{"id":108,"label":862},"2 × 3² × 7",[105],[865,866],"Every factor in the answer must be prime.","84 ÷ 2 = 42, 42 ÷ 2 = 21, 21 = 3 × 7.",{"correct":868,"incorrect":869},"84 = 2 × 2 × 3 × 7 = 2² × 3 × 7.","84 = 2² × 3 × 7. Option (a) and (b) contain the composites 42 and 4, and (d) multiplies to 126.",{"id":871,"type":53,"title":872,"eyebrow":873,"navLabel":874},"ch11","Common mix-ups and how to avoid them","Chapter 11","11 Mix-ups",{"id":876,"type":47,"variant":877,"title":878,"markdown":879},"misc-list","misconception","Six mix-ups to watch for","1. **\"1 is prime.\"** It has one factor only: neither prime nor composite.\n2. **\"All odd numbers are prime.\"** There are 25 odd composites up to 100: 9, 15, 21, 25, 27, 33, 35, 39, 45, 49, 51, 55, 57, 63, 65, 69, 75, 77, 81, 85, 87, 91, 93, 95 and 99.\n3. **\"Co-prime means both numbers are prime.\"** 8 and 15 are co-prime.\n4. **\"To test for prime, try every number up to half of it.\"** You only need primes up to its square root.\n5. **\"A factor tree for 24 like 24 → 4 × 6 is finished.\"** Not until every leaf is prime: 2 × 2 × 2 × 3.\n6. **\"Divisible by 2 and 4 means divisible by 8.\"** Only co-prime pieces combine: 12 is a counterexample.",{"id":881,"type":47,"variant":87,"title":882,"markdown":883},"nuance-big","Big numbers can be prime too","It is tempting to think large numbers are \"too big\" to be prime. But primes never run out: there are primes with millions of digits. The largest known ones are found by computers running for weeks. Large primes are also what keep online banking secure, which you will meet in the Extend layer.",{"id":885,"type":886,"title":887,"terms":888},"gloss-understand","glossary","Vocabulary: precise terms",[889,893,897,901,905,909,913,917,920,924,928,932,936,939,943,947,950],{"term":890,"meaning":891,"example":892},"divisor","Another word for factor: a number that divides another exactly.","6 is a divisor of 42.",{"term":894,"meaning":895,"example":896},"divisible by","Can be divided by a number with remainder 0.","42 is divisible by 7.",{"term":898,"meaning":899,"example":900},"proper factor","A factor of a number other than the number itself.","Proper factors of 10: 1, 2, 5.",{"term":902,"meaning":903,"example":904},"common factor","A number that is a factor of two or more numbers.","3 is a common factor of 12 and 15.",{"term":906,"meaning":907,"example":908},"highest common factor (HCF)","The largest number that is a factor of all the given numbers. Also called the greatest common divisor.","HCF of 12 and 18 is 6.",{"term":910,"meaning":911,"example":912},"common multiple","A number that is a multiple of two or more numbers.","24 is a common multiple of 6 and 8.",{"term":914,"meaning":915,"example":916},"lowest common multiple (LCM)","The smallest number that is a multiple of all the given numbers.","LCM of 4 and 6 is 12.",{"term":918,"meaning":919},"trial division","Testing whether a number is prime by dividing it by primes up to its square root.",{"term":921,"meaning":922,"example":923},"square root","The number that, multiplied by itself, gives the original number. Written √.","√49 = 7",{"term":925,"meaning":926,"example":927},"divisibility rule","A shortcut test for whether a number is divisible by a small number, using its digits.","Digit sum for 3 and 9",{"term":929,"meaning":930,"example":931},"digit sum","The total of a number’s digits.","Digit sum of 2,358 is 18.",{"term":933,"meaning":934,"example":935},"prime factorisation","Writing a number as a product of prime numbers.","60 = 2 × 2 × 3 × 5",{"term":937,"meaning":938},"factor tree","A branching diagram that splits a number into factors until every branch ends in a prime.",{"term":940,"meaning":941,"example":942},"index form","Writing repeated factors with a small raised number (the index or power).","2 × 2 × 2 × 3 = 2³ × 3",{"term":944,"meaning":945,"example":946},"relatively prime","Another name for co-prime: having HCF 1.","9 and 20",{"term":948,"meaning":949},"prime triplet","Here: three primes each 2 apart. The only example is 3, 5, 7.",{"term":951,"meaning":952},"twin prime conjecture","The unproved idea that there are infinitely many twin prime pairs.",{"id":954,"type":955,"title":956,"questions":957},"quiz-understand","quiz","Check your understanding",[958,967,978,990,1003,1011,1024,1037,1046,1059,1072],{"itemId":959,"prompt":960,"options":961,"correct":105,"why":966},"prime-and-composite.understand-q-factors-100","How many factors does 100 have?",[962,963,964,965],{"id":99,"label":161},{"id":102,"label":165},{"id":105,"label":169},{"id":108,"label":173},"1, 2, 4, 5, 10, 20, 25, 50, 100. 100 is a square (10 × 10), so it has an odd number of factors: 9.",{"itemId":968,"prompt":969,"options":970,"correct":102,"why":977},"prime-and-composite.understand-q-stop","To test whether 150 is prime by trial division, what is the largest prime you might need to try?",[971,972,973,975],{"id":99,"label":323},{"id":102,"label":33},{"id":105,"label":974},"13",{"id":108,"label":976},"73","11 × 11 = 121 is at most 150, but 13 × 13 = 169 is more than 150. So stop at 11. (In fact 150 is even, so 2 settles it at once.)",{"itemId":979,"prompt":980,"options":981,"correct":102,"why":989},"prime-and-composite.understand-q-sieve-7","In the sieve up to 100, which numbers does 7 cross out that were not already crossed out?",[982,984,985,987],{"id":99,"label":983},"14, 21, 28",{"id":102,"label":324},{"id":105,"label":986},"7, 49, 91",{"id":108,"label":988},"63, 77, 91","Every other multiple of 7 up to 100 has a smaller prime factor (2, 3 or 5) and is already gone. 7 itself is circled, not crossed out.",{"itemId":991,"prompt":992,"options":993,"correct":105,"why":1002},"prime-and-composite.understand-q-div4","Which number is divisible by 4?",[994,996,998,1000],{"id":99,"label":995},"1,234",{"id":102,"label":997},"5,318",{"id":105,"label":999},"6,732",{"id":108,"label":1001},"9,110","Check the last two digits: 34, 18, 32, 10. Only 32 = 4 × 8.",{"itemId":1004,"prompt":1005,"options":1006,"correct":99,"why":1010},"prime-and-composite.understand-q-div11","Is 3,146 divisible by 11?",[1007,1008],{"id":99,"label":475},{"id":102,"label":1009},"No","From the right: 6 − 4 + 1 − 3 = 0, so yes. Indeed 3,146 = 11 × 286.",{"itemId":1012,"prompt":1013,"options":1014,"correct":105,"why":1023},"prime-and-composite.understand-q-coprime","Which pair is **not** co-prime?",[1015,1017,1019,1021],{"id":99,"label":1016},"15 and 28",{"id":102,"label":1018},"27 and 64",{"id":105,"label":1020},"35 and 42",{"id":108,"label":1022},"11 and 100","35 = 5 × 7 and 42 = 2 × 3 × 7 share the prime 7.",{"itemId":1025,"prompt":1026,"options":1027,"correct":99,"why":1036},"prime-and-composite.understand-q-pf-90","The prime factorisation of 90 is…",[1028,1030,1032,1034],{"id":99,"label":1029},"2 × 3² × 5",{"id":102,"label":1031},"2 × 45",{"id":105,"label":1033},"9 × 10",{"id":108,"label":1035},"2² × 3 × 5","90 = 2 × 45 = 2 × 3 × 3 × 5 = 2 × 3² × 5. (d) multiplies to 60.",{"itemId":1038,"prompt":1039,"options":1040,"correct":102,"why":1045},"prime-and-composite.understand-q-twin-count","How many twin prime pairs are there up to 100?",[1041,1042,1043,1044],{"id":99,"label":161},{"id":102,"label":165},{"id":105,"label":173},{"id":108,"label":320},"(3,5), (5,7), (11,13), (17,19), (29,31), (41,43), (59,61), (71,73).",{"itemId":1047,"prompt":1048,"options":1049,"correct":99,"why":1058},"prime-and-composite.understand-q-6","A number is divisible by 6 exactly when it is…",[1050,1052,1054,1056],{"id":99,"label":1051},"even and divisible by 3",{"id":102,"label":1053},"divisible by 2 and by 4",{"id":105,"label":1055},"ending in 6",{"id":108,"label":1057},"digit sum 6","6 = 2 × 3, and 2 and 3 are co-prime, so both tests together work. 16 ends in 6 but is not divisible by 6.",{"itemId":1060,"prompt":1061,"options":1062,"correct":102,"why":1071},"prime-and-composite.understand-q-why-one","Why is 1 not counted as a prime?",[1063,1065,1067,1069],{"id":99,"label":1064},"It is too small",{"id":102,"label":1066},"It has only one factor, and counting it would spoil unique prime factorisation",{"id":105,"label":1068},"It is odd",{"id":108,"label":1070},"It is a factor of every number","A prime needs exactly two different factors, and leaving 1 out keeps each number’s prime factorisation unique.",{"itemId":1073,"prompt":1074,"options":1075,"correct":102,"why":1083},"prime-and-composite.understand-q-composite-proof","What is enough to prove that 437 is composite?",[1076,1077,1079,1081],{"id":99,"label":1068},{"id":102,"label":1078},"Showing 437 = 19 × 23",{"id":105,"label":1080},"It does not end in 5",{"id":108,"label":1082},"Its digit sum is 14","One factor pair other than 1 × 437 proves it is composite. 19 × 23 = 437.",{"id":1085,"type":1086,"prompt":1087},"reflect-understand","reflection","Explain to a younger student, in your own words, why you can stop at 7 when you sieve the numbers up to 100. Then explain how far you would need to go for the numbers up to 200.",{"id":1089,"type":1090,"title":1091,"points":1092},"cheat-understand","summary","Cheat sheet",[1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103],"**3 × 5 = 15** means: 3 is a factor (divisor) of 15; 15 is a multiple of 3; 15 is divisible by 3.","**Factor-pair method:** try 1, 2, 3, … and stop when the pairs meet. Squares have an odd number of factors.","**Prime:** exactly 2 factors. **Composite:** more than 2 (has a factor other than 1 and itself). **1:** neither.","**2** is the only even prime. The smallest composite is **4**.","**Sieve to 100:** cross out multiples of 2, 3, 5, 7, starting each at p × p. 25 primes remain.","**Trial division:** try primes p while p × p ≤ n. If none divides n, it is prime.","**Rules:** 2, 5, 10 → last digit; 4 → last two digits; 8 → last three; 3, 9 → digit sum; 6 → 2 and 3; 11 → alternating sum.","**Combine rules** only with co-prime pieces: 12 = 3 × 4, 18 = 2 × 9, 24 = 3 × 8.","**Twin primes:** differ by 2. Eight pairs up to 100. 3, 5, 7 is the only prime triplet.","**Co-prime:** HCF 1. Need not be primes (8, 15). Consecutive numbers always are; two evens never are.","**Prime factorisation:** factor tree or division ladder. 360 = 2³ × 3² × 5. Every tree gives the same primes.",{"id":1105,"type":1106,"sourceIds":1107},"sources-understand","sources",[1108,1109,1110,1111,1112,1113],"prime-and-composite-ncert-class6-playing-numbers","prime-and-composite-ncert-class6-prime-time","prime-and-composite-mathisfun-prime","prime-and-composite-mathisfun-divisibility","prime-and-composite-britannica-eratosthenes-sieve","prime-and-composite-wiki-twin-prime",[1108,1109,1110,1111,1112,1113],"needs_review",{"generatedBy":1117,"notes":1118},"claude-code","Draft generated locally with Python-checked numbers; pending owner review.","a1ad883bd6faf7108f932b4ac93737f1f0d569f00cf0873183176c4b0a437546",{"logic:practice":1121,"component:prime-sieve@1":1122,"component:sort-game@1":1123,"component:factor-tree@1":1124,"component:match-pairs@1":1125,"source:prime-and-composite-britannica-eratosthenes-sieve":1126,"source:prime-and-composite-mathisfun-divisibility":1127,"source:prime-and-composite-mathisfun-prime":1128,"source:prime-and-composite-ncert-class6-playing-numbers":1129,"source:prime-and-composite-ncert-class6-prime-time":1130,"source:prime-and-composite-wiki-twin-prime":1131},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","76baccc44f3b2ac333537045fea84801ea3c46f2d75f3db8a02355ef0f267041","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","dfb598a254f20e70550d586121be496f7a546ba0254060fd97b95cb919dfd934","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","af8e7bbdca2e2859b452689299eeef54805896728cef8a361ea1ec0dd63d12d4","d26ef4a1f28e097000ad535bd8d0bb52463c4eb61d815cd4dd383a8db4bac076","ae2cb5b494e894b59fc2f34095d5ba7519d897f66195dc8bef83cbbcffa674b1","d766224fdc533b260af62a8660196dabe3a5e506c901ab99ee8eba647c9a7616","f6edb540745600a1873f0a5851335bf576bd243816f181e62085831fd55ac4ce","b0b179308ab5f683db390f67f15856137356f04cbd838b8129c6b89a3c2b8265",{"state":1133,"reviewer":1134,"selfReview":806,"reviewedAt":1135,"method":1136},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899598633]