[{"data":1,"prerenderedAt":1135},["ShallowReactive",2],{"questions:prime-and-composite":3},{"bank":4,"contentHash":1128,"dependencyHashes":1129,"releaseId":1134},{"schemaVersion":5,"conceptId":6,"revision":5,"title":7,"intro":8,"sections":9,"questions":46,"sourceIds":1120,"reviewStatus":1124,"authoring":1125},1,"prime-and-composite","Prime and composite numbers: question bank","Practise every part of the topic, from finding factors to olympiad-style puzzles. Questions are grouped by skill and graded **foundation**, **core**, **stretch** and **challenge**. Every question has a worked solution: read it even when you get the answer right, because it often shows a faster method.",[10,14,18,22,26,30,34,38,42],{"id":11,"title":12,"description":13},"factors-multiples","Factors and multiples","Listing factors with factor pairs, multiples, and the link between the two.",{"id":15,"title":16,"description":17},"prime-composite","Prime or composite?","Identifying primes and composites, including numbers that look prime but are not.",{"id":19,"title":20,"description":21},"one-and-two","The special cases 1 and 2","Why 1 is neither prime nor composite, and why 2 is the only even prime.",{"id":23,"title":24,"description":25},"sieve-testing","Sieve and primality testing","The Sieve of Eratosthenes and trial division up to the square root.",{"id":27,"title":28,"description":29},"divisibility","Divisibility rules","Tests for 2, 3, 4, 5, 6, 8, 9, 10, 11, missing digits and combining rules.",{"id":31,"title":32,"description":33},"twin-primes","Twin primes and triplets","Twin prime pairs, their patterns, and why 3, 5, 7 is the only prime triplet.",{"id":35,"title":36,"description":37},"co-primes","Co-prime numbers","Pairs with HCF 1, including co-prime composites and consecutive numbers.",{"id":39,"title":40,"description":41},"factorisation","Prime factorisation","Factor trees, index form and counting factors from a factorisation.",{"id":43,"title":44,"description":45},"reasoning","Reasoning and puzzles","Word problems, proofs and olympiad-style puzzles using primes.",[47,61,85,94,113,125,147,166,174,186,197,215,233,247,265,274,296,305,314,323,339,354,365,372,386,403,412,421,439,446,464,481,499,516,525,542,559,577,594,609,618,636,649,657,666,684,695,713,721,729,744,759,769,788,805,817,835,843,861,871,888,897,915,933,952,961,978,987,1005,1014,1022,1040,1050,1060,1070,1078,1087,1095,1105],{"id":48,"section":11,"level":49,"prompt":50,"check":51,"hints":55,"solution":58,"skills":59},"prime-and-composite.q001","foundation","How many factors does **24** have?",{"kind":52,"answer":53,"tolerance":54},"number",8,0,[56,57],"Look for pairs that multiply to 24.","Stop when the pairs meet in the middle.","Find the factor pairs: 1 × 24, 2 × 12, 3 × 8, 4 × 6. 5 does not divide 24, and the next number to try, 6, is already in a pair, so stop. The factors are 1, 2, 3, 4, 6, 8, 12, 24: **8 factors**.",[60],"listing factors",{"id":62,"section":11,"level":49,"prompt":63,"check":64,"hints":80,"solution":82,"skills":83},"prime-and-composite.q002","Which of these is a **multiple of 9**?",{"kind":65,"options":66,"correct":79},"choice",[67,70,73,76],{"id":68,"label":69},"a","3",{"id":71,"label":72},"b","27",{"id":74,"label":75},"c","39",{"id":77,"label":78},"d","91",[71],[81],"A multiple of 9 is in the 9 times table.","27 = 9 × 3, so 27 is in the 9 times table. 3 is a factor of 9, not a multiple. 39 = 9 × 4 + 3 and 91 = 9 × 10 + 1 leave remainders.",[84],"multiples",{"id":86,"section":11,"level":49,"prompt":87,"check":88,"hints":90,"solution":92,"skills":93},"prime-and-composite.q003","What is the **7th multiple** of 8?",{"kind":52,"answer":89,"tolerance":54},56,[91],"The 1st multiple of 8 is 8 itself.","The multiples of 8 are 8 × 1, 8 × 2, 8 × 3, … The 7th is 8 × 7 = **56**.",[84],{"id":95,"section":11,"level":49,"prompt":96,"check":97,"hints":108,"solution":110,"skills":111},"prime-and-composite.q004","Ravi writes: \"The factors of 16 are 1, 2, 4, 6, 8, 16.\" Which number in his list is **wrong**?",{"kind":65,"options":98,"correct":107},[99,101,103,105],{"id":68,"label":100},"1",{"id":71,"label":102},"4",{"id":74,"label":104},"6",{"id":77,"label":106},"16",[74],[109],"Divide 16 by each number in the list.","16 ÷ 6 = 2 remainder 4, so 6 is not a factor. The factor pairs of 16 are 1 × 16, 2 × 8 and 4 × 4, so the factors are 1, 2, 4, 8, 16.",[60,112],"spot the mistake",{"id":114,"section":11,"level":115,"prompt":116,"check":117,"hints":119,"solution":122,"skills":123},"prime-and-composite.q005","core","A sweet shop in Madurai packs **36 laddoos** into boxes with the same number in each box and none left over. Each box must hold **more than 1** laddoo and **fewer than 36**. How many different box sizes are possible?",{"kind":52,"answer":118,"tolerance":54},7,[120,121],"List the factors of 36.","Remove 1 and 36 from the list.","The box size must be a factor of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. Leave out 1 and 36. That leaves 2, 3, 4, 6, 9, 12 and 18: **7 box sizes**.",[60,124],"word problem",{"id":126,"section":11,"level":115,"prompt":127,"check":128,"hints":143,"solution":145,"skills":146},"prime-and-composite.q006","Select **all** the factors of **45**.",{"kind":65,"options":129,"correct":142},[130,131,133,134,136,139],{"id":68,"label":69},{"id":71,"label":132},"5",{"id":74,"label":104},{"id":77,"label":135},"9",{"id":137,"label":138},"e","15",{"id":140,"label":141},"f","18",[68,71,77,137],[144],"An even number can never be a factor of an odd number.","The factors of 45 are 1, 3, 5, 9, 15, 45. From the options, 3, 5, 9 and 15 divide 45 exactly. 6 and 18 are even, and 45 is odd, so they cannot be factors.",[60],{"id":148,"section":11,"level":115,"prompt":149,"check":150,"hints":161,"solution":163,"skills":164},"prime-and-composite.q007","Which statement is **true**?",{"kind":65,"options":151,"correct":160},[152,154,156,158],{"id":68,"label":153},"Every number is a factor of each of its multiples",{"id":71,"label":155},"A number has infinitely many factors",{"id":74,"label":157},"The largest factor of a number is always half of it",{"id":77,"label":159},"A number has only a few multiples",[68],[162],"Think about 4 and its multiples 4, 8, 12, …","If 3 × 4 = 12, then 3 is a factor of 12 and 12 is a multiple of 3. So every number is a factor of each of its multiples. A number has only a **few** factors (at most itself) but **infinitely many** multiples, and its largest factor is the number itself.",[165],"factors vs multiples",{"id":167,"section":11,"level":115,"prompt":168,"check":169,"hints":170,"solution":172,"skills":173},"prime-and-composite.q008","What is the **sum** of all the factors of **28**?",{"kind":52,"answer":89,"tolerance":54},[171],"Pairs: 1 × 28, 2 × 14, 4 × 7.","The factors of 28 are 1, 2, 4, 7, 14, 28. Their sum is 1 + 2 + 4 + 7 + 14 + 28 = **56**. (The proper factors, without 28, add to exactly 28. That makes 28 a *perfect number*.)",[60],{"id":175,"section":11,"level":176,"prompt":177,"check":178,"hints":180,"solution":183,"skills":184},"prime-and-composite.q009","stretch","How many multiples of **7** lie between **100 and 200**?",{"kind":52,"answer":179,"tolerance":54},14,[181,182],"Find the first and last multiples of 7 in the range.","Count the multipliers, not the multiples.","The first multiple of 7 above 100 is 7 × 15 = 105. The last below 200 is 7 × 28 = 196. So the multipliers run from 15 to 28: 28 − 15 + 1 = **14 multiples**.",[84,185],"counting",{"id":187,"section":11,"level":176,"prompt":188,"check":189,"hints":191,"solution":194,"skills":195},"prime-and-composite.q010","What is the **smallest** number that has exactly **6** factors?",{"kind":52,"answer":190,"tolerance":54},12,[192,193],"Try 8, 10, 12, …","Count factor pairs.","Check numbers in order: 1 to 11 all have 4 or fewer factors. 12 has factors 1, 2, 3, 4, 6, 12: exactly six. So the answer is **12**.",[196],"counting factors",{"id":198,"section":15,"level":49,"prompt":199,"check":200,"hints":210,"solution":212,"skills":213},"prime-and-composite.q011","Which of these numbers is **prime**?",{"kind":65,"options":201,"correct":209},[202,204,206,208],{"id":68,"label":203},"21",{"id":71,"label":205},"23",{"id":74,"label":207},"25",{"id":77,"label":72},[71],[211],"Try dividing by 3 and 5.","23 has only the factors 1 and 23. The others are composite: 21 = 3 × 7, 25 = 5 × 5 and 27 = 3 × 9.",[214],"identify primes",{"id":216,"section":15,"level":49,"prompt":217,"check":218,"hints":228,"solution":230,"skills":231},"prime-and-composite.q012","Which of these numbers is **composite**?",{"kind":65,"options":219,"correct":227},[220,222,224,226],{"id":68,"label":221},"2",{"id":71,"label":223},"13",{"id":74,"label":225},"29",{"id":77,"label":75},[77],[229],"Add the digits of each number.","39 = 3 × 13, so it has the factors 1, 3, 13 and 39: more than two. 2, 13 and 29 are prime.",[232],"identify composites",{"id":234,"section":15,"level":49,"prompt":235,"check":236,"hints":243,"solution":244,"skills":245},"prime-and-composite.q013","True or false: **every composite number has at least three factors**.",{"kind":65,"options":237,"correct":242},[238,240],{"id":68,"label":239},"True",{"id":71,"label":241},"False",[68],[],"**True.** A composite number has more than two factors by definition, so at least three. The smallest composite, 4, has exactly three: 1, 2 and 4.",[246],"definitions",{"id":248,"section":15,"level":49,"prompt":249,"check":250,"hints":261,"solution":263,"skills":264},"prime-and-composite.q014","A teacher has **47** students. Can she arrange them in equal rows with **more than one row** and **more than one student in each row**?",{"kind":65,"options":251,"correct":260},[252,254,256,258],{"id":68,"label":253},"Yes: 47 is odd",{"id":71,"label":255},"No: 47 is prime, so its only factor pair is 1 × 47",{"id":74,"label":257},"Yes: 7 rows of 7",{"id":77,"label":259},"No: 47 is composite",[71],[262],"Is 47 in any times table except 1 and 47?","47 is prime: 47 ÷ 2, 3, 5 and 6 all leave remainders, and 7 × 7 = 49 is already more than 47. So the only arrangement is 1 row of 47 (or 47 rows of 1).",[124,214],{"id":266,"section":15,"level":115,"prompt":267,"check":268,"hints":270,"solution":272,"skills":273},"prime-and-composite.q015","How many prime numbers are there **between 30 and 50**?",{"kind":52,"answer":269,"tolerance":54},5,[271],"Check only numbers ending in 1, 3, 7 or 9.","The primes between 30 and 50 are 31, 37, 41, 43 and 47: **5 primes**. 33, 39 and 49 look prime but are 3 × 11, 3 × 13 and 7 × 7.",[214],{"id":275,"section":15,"level":115,"prompt":276,"check":277,"hints":292,"solution":294,"skills":295},"prime-and-composite.q016","Select **all** the prime numbers.",{"kind":65,"options":278,"correct":291},[279,281,283,285,287,289],{"id":68,"label":280},"51",{"id":71,"label":282},"53",{"id":74,"label":284},"57",{"id":77,"label":286},"59",{"id":137,"label":288},"61",{"id":140,"label":290},"63",[71,77,137],[293],"Use the digit-sum test for 3.","53, 59 and 61 are prime. 51 = 3 × 17, 57 = 3 × 19 and 63 = 7 × 9: each has digit sum divisible by 3.",[214],{"id":297,"section":15,"level":115,"prompt":298,"check":299,"hints":301,"solution":303,"skills":304},"prime-and-composite.q017","How many **composite** numbers are there from **1 to 20**?",{"kind":52,"answer":300,"tolerance":54},11,[302],"Count the primes first, and remember 1.","From 1 to 20 there are 8 primes (2, 3, 5, 7, 11, 13, 17, 19) and the number 1, which is neither. So 20 − 8 − 1 = **11** composites: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20.",[185],{"id":306,"section":15,"level":176,"prompt":307,"check":308,"hints":310,"solution":312,"skills":313},"prime-and-composite.q018","What is the **sum** of all the prime numbers less than 20?",{"kind":52,"answer":309,"tolerance":54},77,[311],"There are eight primes below 20.","2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 = **77**.",[214],{"id":315,"section":15,"level":176,"prompt":316,"check":317,"hints":319,"solution":321,"skills":322},"prime-and-composite.q019","What is the smallest **odd** composite number that is **not** divisible by 3?",{"kind":52,"answer":318,"tolerance":54},25,[320],"List the odd composites in order.","Odd composites in order: 9 (3 × 3), 15 (3 × 5), 21 (3 × 7), 25 (5 × 5). The first three are multiples of 3, so the answer is **25**.",[232,43],{"id":324,"section":15,"level":176,"prompt":325,"check":326,"hints":335,"solution":337,"skills":338},"prime-and-composite.q020","Kiran says: \"A number with an **odd number of factors** must be prime.\" Which number proves him wrong?",{"kind":65,"options":327,"correct":334},[328,330,331,333],{"id":68,"label":329},"7",{"id":71,"label":135},{"id":74,"label":332},"12",{"id":77,"label":223},[71],[336],"How many factors does a prime have?","9 has factors 1, 3, 9: three factors, an odd number, but 9 is composite. (Primes always have exactly **two** factors, an even number. Only square numbers have an odd number of factors.)",[112,43],{"id":340,"section":19,"level":49,"prompt":341,"check":342,"hints":351,"solution":352,"skills":353},"prime-and-composite.q021","The number **1** is…",{"kind":65,"options":343,"correct":350},[344,346,348],{"id":68,"label":345},"prime",{"id":71,"label":347},"composite",{"id":74,"label":349},"neither prime nor composite",[74],[],"1 has only **one** factor, itself. A prime needs exactly two and a composite needs more than two, so 1 is **neither**.",[246],{"id":355,"section":19,"level":49,"prompt":356,"check":357,"hints":362,"solution":363,"skills":364},"prime-and-composite.q022","True or false: **2 is the only even prime number**.",{"kind":65,"options":358,"correct":361},[359,360],{"id":68,"label":239},{"id":71,"label":241},[68],[],"**True.** Every even number is divisible by 2. For 2 itself, the factors are just 1 and 2. Every larger even number has 1, 2 and itself as factors, so it is composite.",[246],{"id":366,"section":19,"level":49,"prompt":367,"check":368,"hints":369,"solution":370,"skills":371},"prime-and-composite.q023","How many factors does the number **1** have?",{"kind":52,"answer":5,"tolerance":54},[],"Only one: 1 itself. That is why it is neither prime nor composite.",[246],{"id":373,"section":19,"level":115,"prompt":374,"check":375,"hints":382,"solution":383,"skills":384},"prime-and-composite.q024","Mira says, \"All prime numbers are odd.\" Which number shows she is wrong?",{"kind":65,"options":376,"correct":381},[377,378,379,380],{"id":68,"label":100},{"id":71,"label":221},{"id":74,"label":135},{"id":77,"label":138},[71],[],"2 is prime (factors 1 and 2) and even. 9 and 15 are odd composites, which would disprove a different claim; 1 is not prime at all.",[385],"counterexample",{"id":387,"section":19,"level":115,"prompt":388,"check":389,"hints":400,"solution":401,"skills":402},"prime-and-composite.q025","Why do mathematicians **not** count 1 as a prime number?",{"kind":65,"options":390,"correct":399},[391,393,395,397],{"id":68,"label":392},"Because 1 is too small",{"id":71,"label":394},"Because 1 has only one factor, and counting it would spoil the unique prime factorisation of every number",{"id":74,"label":396},"Because 1 is odd",{"id":77,"label":398},"Because 1 is not a counting number",[71],[],"A prime has exactly two different factors, and 1 has just one. There is also a deep reason: every number has exactly one prime factorisation (12 = 2 × 2 × 3). If 1 were prime, you could add any number of 1s (12 = 2 × 2 × 3 × 1 × 1), and the factorisation would no longer be unique.",[246,43],{"id":404,"section":19,"level":176,"prompt":405,"check":406,"hints":408,"solution":410,"skills":411},"prime-and-composite.q026","p and q are prime numbers and **p + q is odd**. What must one of them be?",{"kind":52,"answer":407,"tolerance":54},2,[409],"Odd + odd = ?","The sum of two odd numbers is even. For p + q to be odd, one of them must be even. The only even prime is **2**.",[43],{"id":413,"section":19,"level":176,"prompt":414,"check":415,"hints":417,"solution":419,"skills":420},"prime-and-composite.q027","p and q are primes with **p + q = 25** and p smaller than q. What is **p × q**?",{"kind":52,"answer":416,"tolerance":54},46,[418],"Use the previous idea: the sum is odd.","25 is odd, so one prime is even: p = 2. Then q = 23, which is prime. So p × q = 2 × 23 = **46**.",[43],{"id":422,"section":23,"level":49,"prompt":423,"check":424,"hints":435,"solution":436,"skills":437},"prime-and-composite.q028","In the Sieve of Eratosthenes, after you circle **2**, what do you do next?",{"kind":65,"options":425,"correct":434},[426,428,430,432],{"id":68,"label":427},"Cross out every multiple of 2 except 2",{"id":71,"label":429},"Cross out every odd number",{"id":74,"label":431},"Circle every even number",{"id":77,"label":433},"Cross out 3",[68],[],"You cross out 4, 6, 8, … (every multiple of 2 except 2 itself), because they all have 2 as an extra factor. Then you move on to the next number not crossed out, which is 3.",[438],"sieve",{"id":440,"section":23,"level":49,"prompt":441,"check":442,"hints":443,"solution":444,"skills":445},"prime-and-composite.q029","How many prime numbers are there from **1 to 100**?",{"kind":52,"answer":318,"tolerance":54},[],"The sieve leaves **25** primes: 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.",[438],{"id":447,"section":23,"level":115,"prompt":448,"check":449,"hints":460,"solution":462,"skills":463},"prime-and-composite.q030","When you sieve the numbers 1 to 100, which numbers does **5** cross out that were **not already** crossed out?",{"kind":65,"options":450,"correct":459},[451,453,455,457],{"id":68,"label":452},"5, 10, 15, 20",{"id":71,"label":454},"25, 35, 55, 65, 85, 95",{"id":74,"label":456},"25, 45, 75",{"id":77,"label":458},"15, 25, 35, 45",[71],[461],"Which multiples of 5 are neither even nor multiples of 3?","Start at 5 × 5 = 25. Multiples of 5 that are even (10, 20, …) went with 2, and 15, 45, 75 went with 3. The new ones are 25, 35, 55, 65, 85 and 95.",[438],{"id":465,"section":23,"level":115,"prompt":466,"check":467,"hints":476,"solution":478,"skills":479},"prime-and-composite.q031","To test whether **150** is prime by trial division, what is the **largest** prime you might need to try?",{"kind":65,"options":468,"correct":475},[469,470,472,473],{"id":68,"label":329},{"id":71,"label":471},"11",{"id":74,"label":223},{"id":77,"label":474},"75",[71],[477],"Find the prime whose square is just below 150.","11 × 11 = 121 is at most 150, but 13 × 13 = 169 is more than 150. So you try primes up to **11**. (In fact 150 is even, so 2 settles it straight away.)",[480],"trial division",{"id":482,"section":23,"level":115,"prompt":483,"check":484,"hints":495,"solution":497,"skills":498},"prime-and-composite.q032","Is **169** prime?",{"kind":65,"options":485,"correct":494},[486,488,490,492],{"id":68,"label":487},"Yes",{"id":71,"label":489},"No: 169 = 13 × 13",{"id":74,"label":491},"No: 169 = 7 × 24 + 1",{"id":77,"label":493},"No: it is divisible by 3",[71],[496],"What is 13 × 13?","169 = 13 × 13, so it is composite, with factors 1, 13 and 169. It passes the tests for 2, 3, 5, 7 and 11, which is why you must go up to √169 = 13.",[480],{"id":500,"section":23,"level":115,"prompt":501,"check":502,"hints":513,"solution":514,"skills":515},"prime-and-composite.q033","Arun tests **187** by dividing by 2, 3, 5 and 7. None works, so he says 187 is prime. What is wrong?",{"kind":65,"options":503,"correct":512},[504,506,508,510],{"id":68,"label":505},"Nothing: 187 is prime",{"id":71,"label":507},"He stopped too early: 187 = 11 × 17",{"id":74,"label":509},"He should have tried 9",{"id":77,"label":511},"He should have tried every number up to 93",[71],[496],"√187 is between 13 and 14 (13 × 13 = 169, 14 × 14 = 196), so he must try 11 and 13 as well. 187 = 11 × 17. (Trying 9 is never needed, and going up to half the number wastes time.)",[480,112],{"id":517,"section":23,"level":176,"prompt":518,"check":519,"hints":520,"solution":522,"skills":523},"prime-and-composite.q034","You sieve the numbers **1 to 120**. What is the **last** prime whose multiples you need to cross out?",{"kind":52,"answer":118,"tolerance":54},[521],"Which prime squared is just over 120?","11 × 11 = 121 is more than 120, so 11 has nothing new to cross out: its multiples up to 120 (22, 33, …, 110) all have smaller prime factors. The last prime needed is **7**.",[438,524],"square root",{"id":526,"section":23,"level":176,"prompt":527,"check":528,"hints":538,"solution":540,"skills":541},"prime-and-composite.q035","Is **391** prime?",{"kind":65,"options":529,"correct":537},[530,531,533,535],{"id":68,"label":487},{"id":71,"label":532},"No: 391 = 17 × 23",{"id":74,"label":534},"No: 391 = 7 × 56",{"id":77,"label":536},"No: 391 = 3 × 131",[71],[539],"Try primes up to 19.","√391 is just under 20 (20 × 20 = 400). 391 is odd, digit sum 13 (not a multiple of 3), not ending in 0 or 5; 391 ÷ 7, 11 and 13 leave remainders; but 17 × 23 = **391**. So it is composite.",[480],{"id":543,"section":23,"level":176,"prompt":544,"check":545,"hints":555,"solution":557,"skills":558},"prime-and-composite.q036","Is **223** prime?",{"kind":65,"options":546,"correct":554},[547,548,550,552],{"id":68,"label":487},{"id":71,"label":549},"No: divisible by 7",{"id":74,"label":551},"No: divisible by 11",{"id":77,"label":553},"No: divisible by 13",[68],[556],"√223 is just under 15.","14 × 14 = 196 and 15 × 15 = 225, so try primes up to 13. 223 is odd; digit sum 7; ends in 3; 7 × 31 = 217 (remainder 6); 11: 3 − 2 + 2 = 3; 13 × 17 = 221 (remainder 2). None divides, so **223 is prime**.",[480],{"id":560,"section":27,"level":49,"prompt":561,"check":562,"hints":573,"solution":574,"skills":575},"prime-and-composite.q037","Which number is divisible by **5**?",{"kind":65,"options":563,"correct":572},[564,566,568,570],{"id":68,"label":565},"1,234",{"id":71,"label":567},"2,345",{"id":74,"label":569},"3,456",{"id":77,"label":571},"4,567",[71],[],"A number is divisible by 5 when its last digit is 0 or 5. Only 2,345 ends in 5.",[576],"divisibility rules",{"id":578,"section":27,"level":49,"prompt":579,"check":580,"hints":590,"solution":592,"skills":593},"prime-and-composite.q038","Which number is divisible by **3**?",{"kind":65,"options":581,"correct":589},[582,583,585,587],{"id":68,"label":565},{"id":71,"label":584},"2,346",{"id":74,"label":586},"4,441",{"id":77,"label":588},"7,001",[71],[591],"Add the digits.","Add the digits: 1,234 → 10, 2,346 → 15, 4,441 → 13, 7,001 → 8. Only 15 is a multiple of 3, so **2,346** is divisible by 3.",[576],{"id":595,"section":27,"level":115,"prompt":596,"check":597,"hints":605,"solution":607,"skills":608},"prime-and-composite.q039","**7,560** is divisible by which of these? Select all.",{"kind":65,"options":598,"correct":604},[599,600,602,603],{"id":68,"label":102},{"id":71,"label":601},"8",{"id":74,"label":135},{"id":77,"label":471},[68,71,74],[606],"Use a different rule for each.","4: last two digits 60 = 4 × 15 ✓. 8: last three digits 560 = 8 × 70 ✓. 9: digit sum 7 + 5 + 6 + 0 = 18 ✓. 11: from the right 0 − 6 + 5 − 7 = −8, not a multiple of 11 ✗. So 4, 8 and 9.",[576],{"id":610,"section":27,"level":115,"prompt":611,"check":612,"hints":613,"solution":615,"skills":616},"prime-and-composite.q040","The four-digit number **52□4** is divisible by 9. What digit goes in the box?",{"kind":52,"answer":118,"tolerance":54},[614],"The digit sum must be a multiple of 9.","Digit sum = 5 + 2 + □ + 4 = 11 + □. The next multiple of 9 after 11 is 18, so □ = 7. Check: 5,274 = 9 × 586. (27 would need □ = 16, which is not a digit.)",[576,617],"missing digit",{"id":619,"section":27,"level":115,"prompt":620,"check":621,"hints":632,"solution":634,"skills":635},"prime-and-composite.q041","Which number is divisible by **11**?",{"kind":65,"options":622,"correct":631},[623,625,627,629],{"id":68,"label":624},"1,331",{"id":71,"label":626},"1,332",{"id":74,"label":628},"2,468",{"id":77,"label":630},"4,589",[68],[633],"Alternate + and − starting from the units digit.","Use the alternating sum from the right. 1,331: 1 − 3 + 3 − 1 = 0 ✓. 1,332: 2 − 3 + 3 − 1 = 1. 2,468: 8 − 6 + 4 − 2 = 4. 4,589: 9 − 8 + 5 − 4 = 2. Only **1,331** (= 11 × 121).",[576],{"id":637,"section":27,"level":115,"prompt":638,"check":639,"hints":644,"solution":646,"skills":647},"prime-and-composite.q042","True or false: **if a number is divisible by 4 and by 6, it is divisible by 24**.",{"kind":65,"options":640,"correct":643},[641,642],{"id":68,"label":239},{"id":71,"label":241},[71],[645],"Try 12.","**False.** 12 is divisible by 4 and by 6 but not by 24. 4 and 6 share the factor 2, so the tests overlap. To test for 24, use 3 and 8, which are co-prime.",[648],"combining rules",{"id":650,"section":27,"level":176,"prompt":651,"check":652,"hints":653,"solution":655,"skills":656},"prime-and-composite.q043","What is the **smallest** digit that can go in the box so that **37□6** is divisible by **6**?",{"kind":52,"answer":407,"tolerance":54},[654],"Divisible by 6 means divisible by 2 and 3.","It ends in 6, so it is even. For 3: digit sum 3 + 7 + □ + 6 = 16 + □ must be a multiple of 3. 16 + 2 = 18 works, and 16 + 0 and 16 + 1 do not. So □ = **2** (3,726 = 6 × 621).",[576,617],{"id":658,"section":27,"level":176,"prompt":659,"check":660,"hints":662,"solution":664,"skills":665},"prime-and-composite.q044","The number **4572□8** is divisible by **8**. How many different digits could go in the box?",{"kind":52,"answer":661,"tolerance":54},3,[663],"Test 208, 218, 228, …","Only the last three digits matter: 2□8 must be divisible by 8. 208 = 8 × 26, 248 = 8 × 31, 288 = 8 × 36; the others (218, 228, 238, 258, 268, 278, 298) are not. So □ can be 0, 4 or 8: **3 digits**.",[576],{"id":667,"section":27,"level":176,"prompt":668,"check":669,"hints":680,"solution":682,"skills":683},"prime-and-composite.q045","Which test correctly checks whether a number is divisible by **12**?",{"kind":65,"options":670,"correct":679},[671,673,675,677],{"id":68,"label":672},"Divisible by 2 and by 6",{"id":71,"label":674},"Divisible by 3 and by 4",{"id":74,"label":676},"Digit sum divisible by 12",{"id":77,"label":678},"Last digit is 2",[71],[681],"The two parts must be co-prime.","12 = 3 × 4 and 3, 4 are co-prime, so passing both tests guarantees divisibility by 12. (a) fails for 18, (c) fails for 39 (digit sum 12) and (d) fails for 22.",[648],{"id":685,"section":27,"level":686,"prompt":687,"check":688,"hints":690,"solution":693,"skills":694},"prime-and-composite.q046","challenge","How many numbers from **1 to 1,000** are divisible by **2 or 5** (or both)?",{"kind":52,"answer":689,"tolerance":54},600,[691,692],"Count multiples of 2, of 5, and of 10.","Do not count the multiples of 10 twice.","Divisible by 2: 500. Divisible by 5: 200. Both 2 and 5 means divisible by 10 (they are co-prime): 100, and these were counted twice. So 500 + 200 − 100 = **600**.",[185,35],{"id":696,"section":31,"level":49,"prompt":697,"check":698,"hints":709,"solution":710,"skills":711},"prime-and-composite.q047","Which pair are **twin primes**?",{"kind":65,"options":699,"correct":708},[700,702,704,706],{"id":68,"label":701},"13 and 15",{"id":71,"label":703},"17 and 19",{"id":74,"label":705},"19 and 23",{"id":77,"label":707},"23 and 25",[71],[],"Twin primes are two primes that differ by 2. 17 and 19 are both prime. 15 = 3 × 5 and 25 = 5 × 5 are composite, and 19 and 23 differ by 4.",[712],"twin primes",{"id":714,"section":31,"level":115,"prompt":715,"check":716,"hints":717,"solution":719,"skills":720},"prime-and-composite.q048","How many pairs of twin primes are there with both numbers **less than 100**?",{"kind":52,"answer":53,"tolerance":54},[718],"Go through the primes up to 100 in order.","(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73): **8 pairs**.",[712],{"id":722,"section":31,"level":115,"prompt":723,"check":724,"hints":726,"solution":727,"skills":728},"prime-and-composite.q049","What is the **smaller** number in the first pair of twin primes **greater than 100**?",{"kind":52,"answer":725,"tolerance":54},101,[],"101 and 103 are both prime (check primes up to 10: 2, 3, 5, 7), so (101, 103) is the first twin pair above 100. The answer is **101**.",[712],{"id":730,"section":31,"level":115,"prompt":731,"check":732,"hints":740,"solution":742,"skills":743},"prime-and-composite.q050","The only set of three primes that are each 2 apart (a prime triplet) is…",{"kind":733,"accept":734},"text",[735,736,737,738,739],"3, 5, 7","3 5 7","357","3,5,7","3 and 5 and 7",[741],"Start with the smallest odd primes.","**3, 5, 7.** For any n, one of n, n + 2 and n + 4 is a multiple of 3, so for all three to be prime that one must be 3 itself.",[712],{"id":745,"section":31,"level":176,"prompt":746,"check":747,"hints":755,"solution":757,"skills":758},"prime-and-composite.q051","For every twin prime pair after (3, 5), the number **between** them is always a multiple of…",{"kind":65,"options":748,"correct":754},[749,750,751,752],{"id":68,"label":102},{"id":71,"label":132},{"id":74,"label":104},{"id":77,"label":753},"10",[74],[756],"Look at 12, 18, 30, 42.","The middle number is even (it lies between two odd numbers). Of three consecutive numbers one is a multiple of 3, and it cannot be either prime, so it is the middle one. Even and a multiple of 3 means a multiple of **6**: 6, 12, 18, 30, 42, 60, 72, …",[712,43],{"id":760,"section":31,"level":176,"prompt":761,"check":762,"hints":764,"solution":766,"skills":767},"prime-and-composite.q052","17 × 19 + 1 is a perfect square. It is the square of which number?",{"kind":52,"answer":763,"tolerance":54},18,[765],"Work out 17 × 19 + 1 and look for a square.","For twin primes p and p + 2: p × (p + 2) + 1 = (p + 1) × (p + 1). So 17 × 19 + 1 = 18 × 18 = 324. The answer is **18**.",[712,768],"patterns",{"id":770,"section":31,"level":686,"prompt":771,"check":772,"hints":783,"solution":785,"skills":786},"prime-and-composite.q053","Why can there be **no** prime triplet n, n + 2, n + 4 other than 3, 5, 7?",{"kind":65,"options":773,"correct":782},[774,776,778,780],{"id":68,"label":775},"Because big numbers are never prime",{"id":71,"label":777},"Because one of the three numbers is always a multiple of 3",{"id":74,"label":779},"Because one of the three is always even",{"id":77,"label":781},"Nobody knows: it is an unsolved problem",[71],[784],"Think about remainders when dividing by 3.","Divide n by 3. If the remainder is 0, n is a multiple of 3. If it is 1, then n + 2 is. If it is 2, then n + 4 is. So one of the three is always a multiple of 3, and it can only be prime if it equals 3. That gives 3, 5, 7 only.",[787,712],"proof",{"id":789,"section":35,"level":49,"prompt":790,"check":791,"hints":802,"solution":803,"skills":804},"prime-and-composite.q054","Which pair of numbers is **co-prime**?",{"kind":65,"options":792,"correct":801},[793,795,797,799],{"id":68,"label":794},"6 and 9",{"id":71,"label":796},"4 and 9",{"id":74,"label":798},"10 and 15",{"id":77,"label":800},"12 and 16",[71],[],"Co-prime means the only common factor is 1. Factors of 4: 1, 2, 4. Factors of 9: 1, 3, 9. Only 1 is shared. The other pairs share 3, 5 and 4.",[35],{"id":806,"section":35,"level":49,"prompt":807,"check":808,"hints":813,"solution":815,"skills":816},"prime-and-composite.q055","True or false: **two co-prime numbers must both be prime**.",{"kind":65,"options":809,"correct":812},[810,811],{"id":68,"label":239},{"id":71,"label":241},[71],[814],"Think of 8 and 15.","**False.** 8 and 15 are co-prime (they share only the factor 1), but neither is prime. Co-prime is about the **pair** sharing no factor.",[35],{"id":818,"section":35,"level":115,"prompt":819,"check":820,"hints":831,"solution":833,"skills":834},"prime-and-composite.q056","Select **all** the co-prime pairs.",{"kind":65,"options":821,"correct":830},[822,824,826,828],{"id":68,"label":823},"14 and 15",{"id":71,"label":825},"21 and 28",{"id":74,"label":827},"25 and 36",{"id":77,"label":829},"33 and 55",[68,74],[832],"Split each number into primes.","14 and 15 are consecutive, so co-prime. 25 = 5² and 36 = 2² × 3² share no prime, so co-prime. 21 and 28 share 7; 33 and 55 share 11.",[35],{"id":836,"section":35,"level":115,"prompt":837,"check":838,"hints":839,"solution":841,"skills":842},"prime-and-composite.q057","How many numbers from **1 to 20** are co-prime with **20**?",{"kind":52,"answer":53,"tolerance":54},[840],"Which primes make up 20?","20 = 2² × 5. Remove every even number and every multiple of 5. What is left: 1, 3, 7, 9, 11, 13, 17, 19: **8 numbers**.",[35,185],{"id":844,"section":35,"level":115,"prompt":845,"check":846,"hints":857,"solution":859,"skills":860},"prime-and-composite.q058","Which of these pairs is **always** co-prime, whatever the numbers?",{"kind":65,"options":847,"correct":856},[848,850,852,854],{"id":68,"label":849},"Two even numbers",{"id":71,"label":851},"Two consecutive numbers",{"id":74,"label":853},"Two odd numbers",{"id":77,"label":855},"Two multiples of 3",[71],[858],"Try 9 and 15 for the odd case.","Any common factor of two consecutive numbers must divide their difference, 1. So the HCF is 1. Two odd numbers can share a factor (9 and 15 share 3).",[35,43],{"id":862,"section":35,"level":176,"prompt":863,"check":864,"hints":866,"solution":868,"skills":869},"prime-and-composite.q059","8 and 15 are co-prime. What is their **lowest common multiple**?",{"kind":52,"answer":865,"tolerance":54},120,[867],"For co-prime numbers, the LCM is the product.","Co-prime numbers share no prime, so their LCM is simply their product: 8 × 15 = **120**.",[35,870],"LCM",{"id":872,"section":35,"level":176,"prompt":873,"check":874,"hints":884,"solution":886,"skills":887},"prime-and-composite.q060","Two numbers are co-prime. One of them is **35**. Which could be the other?",{"kind":65,"options":875,"correct":883},[876,878,879,881],{"id":68,"label":877},"14",{"id":71,"label":207},{"id":74,"label":880},"42",{"id":77,"label":882},"48",[77],[885],"35 = 5 × 7.","35 = 5 × 7. The other number must avoid both 5 and 7. 14 = 2 × 7, 25 = 5², 42 = 2 × 3 × 7 all share a prime with 35. **48** = 2⁴ × 3 shares none.",[35],{"id":889,"section":35,"level":686,"prompt":890,"check":891,"hints":893,"solution":895,"skills":896},"prime-and-composite.q061","Pick two **different** numbers from 1 to 10. Out of the 45 possible pairs, how many are **co-prime**?",{"kind":52,"answer":892,"tolerance":54},31,[894],"Work number by number and only count larger partners.","1 is co-prime with all 9 others (9 pairs). Then count, for each number, the larger partners it is co-prime with: 2 → 3, 5, 7, 9 (4); 3 → 4, 5, 7, 8, 10 (5); 4 → 5, 7, 9 (3); 5 → 6, 7, 8, 9 (4); 6 → 7 (1); 7 → 8, 9, 10 (3); 8 → 9 (1); 9 → 10 (1). Total 9 + 4 + 5 + 3 + 4 + 1 + 3 + 1 + 1 = **31**.",[35,185],{"id":898,"section":39,"level":49,"prompt":899,"check":900,"hints":911,"solution":912,"skills":913},"prime-and-composite.q062","What is the prime factorisation of **12**?",{"kind":65,"options":901,"correct":910},[902,904,906,908],{"id":68,"label":903},"2 × 6",{"id":71,"label":905},"3 × 4",{"id":74,"label":907},"2 × 2 × 3",{"id":77,"label":909},"1 × 12",[74],[],"Every factor must be prime. 12 = 2 × 6 = 2 × 2 × 3. The other options contain the composites 6, 4 or 12, or the number 1.",[914],"prime factorisation",{"id":916,"section":39,"level":49,"prompt":917,"check":918,"hints":929,"solution":930,"skills":931},"prime-and-composite.q063","A factor tree for 60 ends with the leaves **4, 3 and 5**. Why is it not finished?",{"kind":65,"options":919,"correct":928},[920,922,924,926],{"id":68,"label":921},"3 is not prime",{"id":71,"label":923},"4 is not prime and must be split into 2 × 2",{"id":74,"label":925},"5 is not prime",{"id":77,"label":927},"It is finished",[71],[],"A factor tree is finished only when every leaf is prime. 4 = 2 × 2, so the tree continues: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.",[932,112],"factor trees",{"id":934,"section":39,"level":115,"prompt":935,"check":936,"hints":947,"solution":949,"skills":950},"prime-and-composite.q064","Which is the prime factorisation of **72** in index form?",{"kind":65,"options":937,"correct":946},[938,940,942,944],{"id":68,"label":939},"2² × 3³",{"id":71,"label":941},"2³ × 3²",{"id":74,"label":943},"8 × 9",{"id":77,"label":945},"2 × 36",[71],[948],"72 = 8 × 9. Split each.","72 = 8 × 9 = 2 × 2 × 2 × 3 × 3 = **2³ × 3²**. Check the other: 2² × 3³ = 4 × 27 = 108.",[951],"index form",{"id":953,"section":39,"level":115,"prompt":954,"check":955,"hints":957,"solution":959,"skills":960},"prime-and-composite.q065","What number has the prime factorisation **2² × 3 × 5²**?",{"kind":52,"answer":956,"tolerance":54},300,[958],"Work out each power first.","2² × 3 × 5² = 4 × 3 × 25 = **300**.",[951],{"id":962,"section":39,"level":115,"prompt":963,"check":964,"hints":975,"solution":976,"skills":977},"prime-and-composite.q066","What is the prime factorisation of **210**?",{"kind":65,"options":965,"correct":974},[966,968,970,972],{"id":68,"label":967},"2 × 3 × 5 × 7",{"id":71,"label":969},"2 × 105",{"id":74,"label":971},"2 × 3 × 35",{"id":77,"label":973},"2² × 3 × 5 × 7",[68],[],"210 = 2 × 105 = 2 × 3 × 35 = 2 × 3 × 5 × 7. It is the product of the first four primes. (d) multiplies to 420.",[914],{"id":979,"section":39,"level":176,"prompt":980,"check":981,"hints":983,"solution":985,"skills":986},"prime-and-composite.q067","540 = 2² × 3³ × 5. How many factors does 540 have?",{"kind":52,"answer":982,"tolerance":54},24,[984],"Add 1 to each power.","Add 1 to each power and multiply: (2 + 1)(3 + 1)(1 + 1) = 3 × 4 × 2 = **24**. Each factor is 2ᵃ × 3ᵇ × 5ᶜ with 3 choices for a, 4 for b and 2 for c.",[196],{"id":988,"section":39,"level":176,"prompt":989,"check":990,"hints":1000,"solution":1002,"skills":1003},"prime-and-composite.q068","Which of these is a **perfect square**?",{"kind":65,"options":991,"correct":999},[992,993,995,997],{"id":68,"label":941},{"id":71,"label":994},"2² × 3² × 5²",{"id":74,"label":996},"2 × 3 × 5",{"id":77,"label":998},"2⁴ × 3",[71],[1001],"Look for all even powers.","A perfect square has every power even. 2² × 3² × 5² = (2 × 3 × 5)² = 30² = 900.",[951,1004],"squares",{"id":1006,"section":39,"level":686,"prompt":1007,"check":1008,"hints":1010,"solution":1012,"skills":1013},"prime-and-composite.q069","What is the **smallest** number that has exactly **9** factors?",{"kind":52,"answer":1009,"tolerance":54},36,[1011],"Think about the shape of the powers: (a + 1)(b + 1) = 9.","9 = 9 or 3 × 3. So the number is p⁸ (smallest 2⁸ = 256) or p² × q² (smallest 2² × 3² = 36). The answer is **36**, with factors 1, 2, 3, 4, 6, 9, 12, 18, 36.",[196],{"id":1015,"section":43,"level":115,"prompt":1016,"check":1017,"hints":1018,"solution":1020,"skills":1021},"prime-and-composite.q070","**91** students must be split into equal teams, each with more than 1 and fewer than 91 students. How many different team sizes are possible?",{"kind":52,"answer":407,"tolerance":54},[1019],"Is 91 divisible by 7?","91 = 7 × 13, so its factors are 1, 7, 13 and 91. Leaving out 1 and 91, the team sizes are 7 or 13: **2 options**. (Many people think 91 is prime, but it is not.)",[124],{"id":1023,"section":43,"level":115,"prompt":1024,"check":1025,"hints":1036,"solution":1038,"skills":1039},"prime-and-composite.q071","Which statement is **always** true?",{"kind":65,"options":1026,"correct":1035},[1027,1029,1031,1033],{"id":68,"label":1028},"The sum of two primes is even",{"id":71,"label":1030},"The product of two primes is composite",{"id":74,"label":1032},"Adding 1 to a prime gives a composite",{"id":77,"label":1034},"Every odd number is prime",[71],[1037],"Test each statement with the prime 2.","p × q always has at least the factors 1, p, q and p × q (or 1, p, p² when p = q), so it is composite. 2 + 3 = 5 is odd, 2 + 1 = 3 is prime and 9 is odd but composite.",[43],{"id":1041,"section":43,"level":176,"prompt":1042,"check":1043,"hints":1046,"solution":1048,"skills":1049},"prime-and-composite.q072","A 13-year cicada brood and a 17-year brood in the same forest both emerge this year. After how many years will they next emerge together?",{"kind":52,"answer":1044,"tolerance":54,"unit":1045},221,"years",[1047],"Find the LCM of 13 and 17.","13 and 17 are different primes, so they are co-prime and their LCM is 13 × 17 = **221 years**.",[870,124],{"id":1051,"section":43,"level":176,"prompt":1052,"check":1053,"hints":1055,"solution":1057,"skills":1058},"prime-and-composite.q073","In how many ways can **50** be written as the sum of two primes? (3 + 47 and 47 + 3 count as the same.)",{"kind":52,"answer":1054,"tolerance":54},4,[1056],"Try each prime up to 25.","3 + 47, 7 + 43, 13 + 37 and 19 + 31: **4 ways**. (5 + 45, 11 + 39, 17 + 33 and 23 + 27 fail because the second number is composite.)",[1059,214],"Goldbach",{"id":1061,"section":43,"level":176,"prompt":1062,"check":1063,"hints":1065,"solution":1067,"skills":1068},"prime-and-composite.q074","A two-digit prime becomes a **different** prime when its digits are reversed, and its digits add up to **10**. What is the smaller of the two primes?",{"kind":52,"answer":1064,"tolerance":54},37,[1066],"List the two-digit numbers whose digits add to 10.","Two-digit numbers with digit sum 10: 19, 28, 37, 46, 55, 64, 73, 82, 91. The primes are 19, 37 and 73. 19 reversed is 91 = 7 × 13, not prime. 37 reversed is 73, prime. So the pair is 37 and 73; the smaller is **37**.",[214,1069],"puzzle",{"id":1071,"section":43,"level":686,"prompt":1072,"check":1073,"hints":1074,"solution":1075,"skills":1076},"prime-and-composite.q075","Find the prime p such that **p, p + 10 and p + 14 are all prime**.",{"kind":52,"answer":661,"tolerance":54},[784],"Look at remainders on division by 3. If p leaves 1, then p + 14 leaves 0 and is a multiple of 3. If p leaves 2, then p + 10 leaves 0. So p must leave 0, i.e. p = 3. Check: 3, 13 and 17 are all prime. The answer is **p = 3**.",[787,1077],"olympiad",{"id":1079,"section":43,"level":686,"prompt":1080,"check":1081,"hints":1082,"solution":1085,"skills":1086},"prime-and-composite.q076","How many zeros are there at the end of **50!** (that is, 1 × 2 × 3 × … × 50)?",{"kind":52,"answer":190,"tolerance":54},[1083,1084],"Count factors of 5.","Remember 25 = 5 × 5.","Each final zero needs a factor 10 = 2 × 5, and there are plenty of 2s, so count the 5s. Multiples of 5 up to 50: 10 of them. Multiples of 25 (25 and 50) give an extra 5 each: 2 more. Total **12 zeros**.",[914,1077],{"id":1088,"section":43,"level":686,"prompt":1089,"check":1090,"hints":1091,"solution":1093,"skills":1094},"prime-and-composite.q077","A corridor has **50** closed lockers. Student 1 opens all of them, student 2 toggles every 2nd locker, student 3 every 3rd, and so on up to student 50. How many lockers end **open**?",{"kind":52,"answer":118,"tolerance":54},[1092],"Which numbers have an odd number of factors?","Locker n is toggled once for each of its factors. It ends open if it is toggled an odd number of times, which happens only for perfect squares: 1, 4, 9, 16, 25, 36, 49. **7 lockers**.",[196,1077],{"id":1096,"section":43,"level":686,"prompt":1097,"check":1098,"hints":1100,"solution":1102,"skills":1103},"prime-and-composite.q078","2 × 3 × 5 × 7 × 11 × 13 + 1 = 30,031. What is the **smallest prime factor** of 30,031?",{"kind":52,"answer":1099,"tolerance":54},59,[1101],"None of 2 to 13 can divide it. Why?","By Euclid’s argument, none of 2, 3, 5, 7, 11, 13 divides 30,031 (each leaves remainder 1). Testing the next primes: 17, 19, 23, 29, 31, 37, 41, 43, 47 and 53 all fail, and 59 × 509 = 30,031. So the smallest prime factor is **59**. This shows Euclid’s number need not be prime, but its prime factors are new.",[1104,480],"Euclid",{"id":1106,"section":43,"level":686,"prompt":1107,"check":1108,"hints":1116,"solution":1118,"skills":1119},"prime-and-composite.q079","For every prime p **greater than 3**, p × p − 1 is divisible by which of these? (Choose the largest that always works.)",{"kind":65,"options":1109,"correct":1115},[1110,1111,1112,1114],{"id":68,"label":104},{"id":71,"label":332},{"id":74,"label":1113},"24",{"id":77,"label":882},[74],[1117],"Factorise p × p − 1 as (p − 1)(p + 1).","p × p − 1 = (p − 1)(p + 1). These are consecutive even numbers, so one is a multiple of 4 and the product has a factor 8. One of p − 1, p, p + 1 is a multiple of 3, and it is not p. So the product is divisible by 8 × 3 = **24**. It is not always divisible by 48: 5 × 5 − 1 = 24.",[787,1077],[1121,1122,1123],"prime-and-composite-ncert-class6-playing-numbers","prime-and-composite-ncert-class6-prime-time","prime-and-composite-mathisfun-divisibility","needs_review",{"generatedBy":1126,"notes":1127},"claude-code","Draft; every numeric answer computed and asserted in Python. Pending owner review.","011567685c802b46257709d7e50e71ee57c0a2c1e5e780fa544883b084ca7f84",{"logic:questions":1130,"source:prime-and-composite-mathisfun-divisibility":1131,"source:prime-and-composite-ncert-class6-playing-numbers":1132,"source:prime-and-composite-ncert-class6-prime-time":1133},"e7fd7c240a65bea1bff6277f7af7cca65e2c7fcb13c6d756cb943d53f3cbc948","d26ef4a1f28e097000ad535bd8d0bb52463c4eb61d815cd4dd383a8db4bac076","d766224fdc533b260af62a8660196dabe3a5e506c901ab99ee8eba647c9a7616","f6edb540745600a1873f0a5851335bf576bd243816f181e62085831fd55ac4ce","preview-7e1cbbcc4f",1789899599980]