[{"data":1,"prerenderedAt":1215},["ShallowReactive",2],{"questions:properties-of-numbers":3},{"bank":4,"contentHash":1201,"dependencyHashes":1202,"releaseId":1214},{"schemaVersion":5,"conceptId":6,"revision":5,"title":7,"intro":8,"sections":9,"questions":42,"sourceIds":1186,"reviewStatus":1197,"authoring":1198},1,"properties-of-numbers","Properties of numbers: question bank","Practise every property of whole numbers: closure, commutative, associative and distributive laws, the special roles of 0 and 1, even and odd rules, property-powered mental maths, and \"always, sometimes or never\" reasoning. Questions go from **foundation** to **challenge**. Every question has a worked solution; try it first, then read the solution even if you were right, because it often shows a faster route.",[10,14,18,22,26,30,34,38],{"id":11,"title":12,"description":13},"closure","Closure","Natural and whole numbers, and whether sums, differences, products and quotients stay in a family.",{"id":15,"title":16,"description":17},"commutative","Commutative property","Swapping the order in addition and multiplication; why subtraction and division fail.",{"id":19,"title":20,"description":21},"associative","Associative property","Choosing which pair to do first; grouping gaps for subtraction and division.",{"id":23,"title":24,"description":25},"distributive","Distributive property","Breaking apart, area models, common factors and special products.",{"id":27,"title":28,"description":29},"identity-zero","Identity, zero and one","Additive and multiplicative identities, the properties of 0 and 1, and why division by zero is undefined.",{"id":31,"title":32,"description":33},"even-odd","Even and odd","Parity rules for sums, differences and products, with reasons and invariant puzzles.",{"id":35,"title":36,"description":37},"mental-maths","Mental maths with properties","Compensation, doubling and halving, × 5, × 11, × 25, × 99, × 101, × 125 and friendly pairs.",{"id":39,"title":40,"description":41},"reasoning","Reasoning and counterexamples","Always, sometimes or never true; counterexamples; proofs; invented operations.",[43,68,82,98,118,137,156,176,193,211,223,235,249,258,276,284,302,313,322,340,350,359,378,395,406,424,433,443,452,461,472,481,500,511,519,537,546,556,565,573,580,595,613,621,640,661,678,686,703,720,738,752,763,776,794,803,821,837,855,871,879,888,897,907,917,928,936,945,955,963,972,982,991,1000,1017,1034,1047,1059,1076,1095,1113,1131,1141,1159,1177],{"id":44,"section":11,"level":45,"prompt":46,"check":47,"hints":63,"solution":65,"skills":66},"properties-of-numbers.q001","foundation","Which of these is **not** a whole number?",{"kind":48,"options":49,"correct":62},"choice",[50,53,56,59],{"id":51,"label":52},"a","0",{"id":54,"label":55},"b","15",{"id":57,"label":58},"c","7 − 9",{"id":60,"label":61},"d","9 − 7",[57],[64],"Whole numbers are 0, 1, 2, 3, …","7 − 9 would need you to go 2 steps to the left of 0 on the number line, which is not a whole number. 0, 15 and 9 − 7 = 2 are all whole numbers.",[67],"whole numbers",{"id":69,"section":11,"level":45,"prompt":70,"check":71,"hints":78,"solution":80,"skills":81},"properties-of-numbers.q002","True or false: the sum of any two whole numbers is always a whole number.",{"kind":48,"options":72,"correct":77},[73,75],{"id":51,"label":74},"True",{"id":54,"label":76},"False",[51],[79],"Can adding ever take you below 0 or between two whole numbers?","True. Whole numbers are **closed under addition**: adding b to a means moving b steps right from a on the number line, and you always land on a whole number.",[11],{"id":83,"section":11,"level":45,"prompt":84,"check":85,"hints":95,"solution":96,"skills":97},"properties-of-numbers.q003","Which is the smallest whole number?",{"kind":48,"options":86,"correct":94},[87,89,90,92],{"id":51,"label":88},"1",{"id":54,"label":52},{"id":57,"label":91},"−1",{"id":60,"label":93},"There is none",[54],[],"Whole numbers are 0, 1, 2, 3, … so the smallest is 0. The smallest natural number is 1.",[67],{"id":99,"section":11,"level":100,"prompt":101,"check":102,"hints":113,"solution":115,"skills":116},"properties-of-numbers.q004","core","Which example shows that whole numbers are **not** closed under division?",{"kind":48,"options":103,"correct":112},[104,106,108,110],{"id":51,"label":105},"12 ÷ 4 = 3",{"id":54,"label":107},"0 ÷ 7 = 0",{"id":57,"label":109},"10 ÷ 4 = 2½",{"id":60,"label":111},"9 ÷ 9 = 1",[57],[114],"Look for an answer that is not a whole number.","10 ÷ 4 = 2½ is not a whole number, so dividing two whole numbers can leave the set. One counterexample is enough to show a set is not closed.",[11,117],"counterexample",{"id":119,"section":11,"level":100,"prompt":120,"check":121,"hints":132,"solution":134,"skills":135},"properties-of-numbers.q005","Natural numbers are closed under which operations? (Choose all that apply.)",{"kind":48,"options":122,"correct":131},[123,125,127,129],{"id":51,"label":124},"Addition",{"id":54,"label":126},"Subtraction",{"id":57,"label":128},"Multiplication",{"id":60,"label":130},"Division",[51,57],[133],"Remember 0 is not a natural number.","Adding or multiplying counting numbers always gives a counting number. Subtraction fails: 5 − 5 = 0 and 3 − 8 are not natural numbers. Division fails: 3 ÷ 4 is not a natural number.",[11,136],"natural numbers",{"id":138,"section":11,"level":100,"prompt":139,"check":140,"hints":151,"solution":153,"skills":154},"properties-of-numbers.q006","Which set is closed under **multiplication** but **not** under addition?",{"kind":48,"options":141,"correct":150},[142,144,146,148],{"id":51,"label":143},"Even numbers",{"id":54,"label":145},"Odd numbers",{"id":57,"label":147},"Multiples of 10",{"id":60,"label":149},"Whole numbers",[54],[152],"Test 3 + 5 and 3 × 5.","odd × odd = odd, so odd numbers are closed under multiplication. But odd + odd = even (3 + 5 = 8), so they are not closed under addition. Even numbers, multiples of 10 and whole numbers are closed under both.",[11,155],"parity",{"id":157,"section":11,"level":158,"prompt":159,"check":160,"hints":171,"solution":173,"skills":174},"properties-of-numbers.q007","stretch","Is the set of **square numbers** (1, 4, 9, 16, …) closed under multiplication?",{"kind":48,"options":161,"correct":170},[162,164,166,168],{"id":51,"label":163},"Yes, because (a × a) × (b × b) = (a × b) × (a × b)",{"id":54,"label":165},"No, because 4 × 9 = 36 is not a square",{"id":57,"label":167},"No, because 1 + 4 = 5 is not a square",{"id":60,"label":169},"Only for even squares",[51],[172],"Try 4 × 9.","Rearranging with the commutative and associative laws, (a × a) × (b × b) = (a × b) × (a × b), which is a square. For example 4 × 9 = 36 = 6 × 6. (Squares are not closed under addition, but that is a different operation.)",[11,175],"proof",{"id":177,"section":11,"level":158,"prompt":178,"check":179,"hints":188,"solution":190,"skills":191},"properties-of-numbers.q008","Which family is closed under **subtraction**?",{"kind":48,"options":180,"correct":187},[181,183,184,186],{"id":51,"label":182},"Natural numbers",{"id":54,"label":149},{"id":57,"label":185},"Integers",{"id":60,"label":145},[57],[189],"Which family includes negative numbers?","Any integer minus any integer is an integer (for example 3 − 8 = −5). Natural and whole numbers fail with 3 − 8; odd numbers fail with 7 − 3 = 4.",[11,192],"integers",{"id":194,"section":15,"level":45,"prompt":195,"check":196,"hints":207,"solution":209,"skills":210},"properties-of-numbers.q009","Which shows the commutative property of addition?",{"kind":48,"options":197,"correct":206},[198,200,202,204],{"id":51,"label":199},"6 + 9 = 9 + 6",{"id":54,"label":201},"(6 + 9) + 1 = 6 + (9 + 1)",{"id":57,"label":203},"6 + 0 = 6",{"id":60,"label":205},"6 × 9 = 9 × 6",[51],[208],"Look for the two numbers in swapped order with a + sign.","Commutative means the two numbers swap places and the answer stays the same. 6 × 9 = 9 × 6 is commutative too, but for multiplication, not addition.",[15],{"id":212,"section":15,"level":45,"prompt":213,"check":214,"hints":218,"solution":220,"skills":221},"properties-of-numbers.q010","Riya sets out 8 rows of 7 chairs. Arjun turns the arrangement round to 7 rows of how many chairs, keeping the same total?",{"kind":215,"answer":216,"tolerance":217},"number",8,0,[219],"Rows and columns swap.","Turning the array round swaps rows and columns: 8 × 7 = 7 × 8 = 56. So there are 7 rows of **8** chairs.",[15,222],"arrays",{"id":224,"section":15,"level":45,"prompt":225,"check":226,"hints":231,"solution":232,"skills":233},"properties-of-numbers.q011","True or false: 15 − 6 = 6 − 15.",{"kind":48,"options":227,"correct":230},[228,229],{"id":51,"label":74},{"id":54,"label":76},[54],[],"False. 15 − 6 = 9, but 6 − 15 is not even a whole number (with integers it is −9). Subtraction is not commutative.",[15,234],"subtraction",{"id":236,"section":15,"level":100,"prompt":237,"check":238,"hints":245,"solution":247,"skills":248},"properties-of-numbers.q012","For which operations is a ∗ b = b ∗ a true for all whole numbers? (Choose all that apply.)",{"kind":48,"options":239,"correct":244},[240,241,242,243],{"id":51,"label":124},{"id":54,"label":126},{"id":57,"label":128},{"id":60,"label":130},[51,57],[246],"Try 8 and 2 in each operation.","Only + and × are commutative. Counterexamples: 5 − 2 = 3 but 2 − 5 is not 3; 8 ÷ 2 = 4 but 2 ÷ 8 = ¼.",[15],{"id":250,"section":15,"level":100,"prompt":251,"check":252,"hints":254,"solution":256,"skills":257},"properties-of-numbers.q013","If 347 × 26 = 9,022, what is 26 × 347?",{"kind":215,"answer":253,"tolerance":217},9022,[255],"Order does not change a product.","By the commutative property of multiplication, 26 × 347 = 347 × 26 = **9,022**. No calculation is needed.",[15],{"id":259,"section":15,"level":100,"prompt":260,"check":261,"hints":272,"solution":274,"skills":275},"properties-of-numbers.q014","For which whole numbers a and b is a − b = b − a?",{"kind":48,"options":262,"correct":271},[263,265,267,269],{"id":51,"label":264},"Never",{"id":54,"label":266},"Only when a = b",{"id":57,"label":268},"Only when b = 0",{"id":60,"label":270},"Always",[54],[273],"Try a = b = 5.","a − b and b − a are both whole numbers only when a = b, and then both are 0. For any other pair one of them is not a whole number (with integers, they are opposites such as 4 and −4).",[15,39],{"id":277,"section":15,"level":100,"prompt":278,"check":279,"hints":281,"solution":282,"skills":283},"properties-of-numbers.q015","Count on from the bigger number: 4 + 89 = ?",{"kind":215,"answer":280,"tolerance":217},93,[],"Turn it round to 89 + 4 and count on 4: 90, 91, 92, **93**. Addition is commutative, so the answer is the same.",[15],{"id":285,"section":15,"level":158,"prompt":286,"check":287,"hints":298,"solution":299,"skills":300},"properties-of-numbers.q016","Kabir says: \"Since 6 × 4 = 4 × 6, a box of 6 packets of 4 pens is the same as 4 packets of 6 pens.\" What is right and wrong with his claim?",{"kind":48,"options":288,"correct":297},[289,291,293,295],{"id":51,"label":290},"Completely right",{"id":54,"label":292},"Same number of pens (24), but not the same packing",{"id":57,"label":294},"Wrong: 6 × 4 is not 4 × 6",{"id":60,"label":296},"Wrong: they differ by 2 pens",[54],[],"The commutative property says the **number** of pens is equal: 24 either way. But 6 packets and 4 packets are different arrangements. Properties tell you about the numbers, not about everything in the situation.",[15,301],"real life",{"id":303,"section":19,"level":45,"prompt":304,"check":305,"hints":308,"solution":310,"skills":311},"properties-of-numbers.q017","At a kirana shop: rice ₹38, dal ₹45, oil ₹55. Add the friendly pair first. What is the total?",{"kind":215,"answer":306,"tolerance":217,"unit":307},138,"₹",[309],"Which two numbers make 100?","45 + 55 = 100 first, then 100 + 38 = **138**. The associative (and commutative) property lets you choose the pair. 38 + 45 + 55 = 138.",[19,312],"mental maths",{"id":314,"section":19,"level":45,"prompt":315,"check":316,"hints":318,"solution":320,"skills":321},"properties-of-numbers.q018","Work out **2 × 13 × 5** in your head.",{"kind":215,"answer":317,"tolerance":217},130,[319],"Which pair makes 10?","Group 2 × 5 = 10 first, then 10 × 13 = **130**. 2 × 13 × 5 = 130.",[19],{"id":323,"section":19,"level":100,"prompt":324,"check":325,"hints":336,"solution":338,"skills":339},"properties-of-numbers.q019","Which equation shows the associative property?",{"kind":48,"options":326,"correct":335},[327,329,331,333],{"id":51,"label":328},"(4 × 5) × 3 = 4 × (5 × 3)",{"id":54,"label":330},"4 × 5 = 5 × 4",{"id":57,"label":332},"4 × (5 + 3) = 20 + 12",{"id":60,"label":334},"4 × 1 = 4",[51],[337],"Look for moved brackets.","Associative: the numbers stay in the same order, only the brackets (which pair is done first) change.",[19],{"id":341,"section":19,"level":100,"prompt":342,"check":343,"hints":345,"solution":348,"skills":349},"properties-of-numbers.q020","How much smaller is (60 − 20) − 10 than 60 − (20 − 10)?",{"kind":215,"answer":344,"tolerance":217},20,[346,347],"Work out both groupings.","The gap is always 2 × the last number.","(60 − 20) − 10 = 30, and 60 − (20 − 10) = 50. The left grouping is 50 − 30 = 20 smaller, which is 2 × 10. Subtraction is not associative.",[19,234],{"id":351,"section":19,"level":100,"prompt":352,"check":353,"hints":355,"solution":357,"skills":358},"properties-of-numbers.q021","Use grouping: **25 × 17 × 4** = ?",{"kind":215,"answer":354,"tolerance":217},1700,[356],"25 × 4 = 100.","Rearrange and group: (25 × 4) × 17 = 100 × 17 = **1,700**. 25 × 17 × 4 = 1,700.",[19,15],{"id":360,"section":19,"level":100,"prompt":361,"check":362,"hints":373,"solution":375,"skills":376},"properties-of-numbers.q022","Which is true?",{"kind":48,"options":363,"correct":372},[364,366,368,370],{"id":51,"label":365},"(48 ÷ 4) ÷ 2 = 48 ÷ (4 ÷ 2)",{"id":54,"label":367},"(48 ÷ 4) ÷ 2 = 6 and 48 ÷ (4 ÷ 2) = 24",{"id":57,"label":369},"Both groupings give 24",{"id":60,"label":371},"Both groupings give 6",[54],[374],"Do the brackets first in each.","(48 ÷ 4) ÷ 2 = 12 ÷ 2 = 6. 48 ÷ (4 ÷ 2) = 48 ÷ 2 = 24. Division is not associative, which is why we divide from left to right.",[19,377],"division",{"id":379,"section":19,"level":100,"prompt":380,"check":381,"hints":391,"solution":392,"skills":393},"properties-of-numbers.q023","What does 50 − 20 − 10 equal, using the agreed left-to-right rule?",{"kind":48,"options":382,"correct":390},[383,385,387,388],{"id":51,"label":384},"20",{"id":54,"label":386},"40",{"id":57,"label":52},{"id":60,"label":389},"60",[51],[],"Work from left to right: 50 − 20 = 30, then 30 − 10 = 20. (50 − (20 − 10) = 40 is a different calculation.)",[19,394],"order of operations",{"id":396,"section":19,"level":158,"prompt":397,"check":398,"hints":401,"solution":403,"skills":404},"properties-of-numbers.q024","A bookshop has 4 shelves. Each shelf holds 25 boxes, and each box holds 36 books. How many books? Use the easiest grouping.",{"kind":215,"answer":399,"tolerance":217,"unit":400},3600,"books",[402],"Multiply 4 × 25 first.","Total = 4 × 25 × 36. Group (4 × 25) × 36 = 100 × 36 = **3,600** books. 4 × 25 × 36 = 3,600.",[19,405],"word problem",{"id":407,"section":19,"level":408,"prompt":409,"check":410,"hints":419,"solution":422,"skills":423},"properties-of-numbers.q025","challenge","For whole numbers with a ≠ 0, when is (a ÷ b) ÷ c equal to a ÷ (b ÷ c)?",{"kind":48,"options":411,"correct":418},[412,413,415,417],{"id":51,"label":270},{"id":54,"label":414},"Only when c = 1",{"id":57,"label":416},"Only when b = c",{"id":60,"label":264},[54],[420,421],"Compare (72 ÷ 6) ÷ 3 with 72 ÷ (6 ÷ 3).","What is the ratio of the two answers?","a ÷ (b ÷ c) is c × c times (a ÷ b) ÷ c. For example (72 ÷ 6) ÷ 3 = 4 and 72 ÷ (6 ÷ 3) = 36 = 9 × 4. They are equal only when c × c = 1, that is c = 1.",[19,39],{"id":425,"section":23,"level":45,"prompt":426,"check":427,"hints":429,"solution":431,"skills":432},"properties-of-numbers.q026","Break apart to multiply: **6 × 14** = 6 × 10 + 6 × 4 = ?",{"kind":215,"answer":428,"tolerance":217},84,[430],"6 × 10 = 60 and 6 × 4 = 24.","6 × 10 + 6 × 4 = 60 + 24 = 84. The 6 multiplies both parts of 14.",[23],{"id":434,"section":23,"level":45,"prompt":435,"check":436,"hints":438,"solution":440,"skills":441},"properties-of-numbers.q027","A pen costs ₹99. What do 4 pens cost? (Think 4 × 100 − 4.)",{"kind":215,"answer":437,"tolerance":217,"unit":307},396,[439],"Pretend each pen is ₹100, then give back ₹1 per pen.","4 × 99 = 4 × 100 − 4 × 1 = 400 − 4 = 396. Answer: **₹396**.",[23,442],"money",{"id":444,"section":23,"level":100,"prompt":445,"check":446,"hints":448,"solution":450,"skills":451},"properties-of-numbers.q028","Use the distributive property: **98 × 25** = ?",{"kind":215,"answer":447,"tolerance":217},2450,[449],"Write 98 as 100 − 2.","98 = 100 − 2, so 98 × 25 = 100 × 25 − 2 × 25 = 2,500 − 50 = **2,450**. 98 × 25 = 2,450.",[23,312],{"id":453,"section":23,"level":100,"prompt":454,"check":455,"hints":457,"solution":459,"skills":460},"properties-of-numbers.q029","Use the distributive property: **12 × 105** = ?",{"kind":215,"answer":456,"tolerance":217},1260,[458],"Write 105 as 100 + 5.","105 = 100 + 5, so 12 × 105 = 1,200 + 60 = **1,260**. 12 × 105 = 1,260.",[23,312],{"id":462,"section":23,"level":100,"prompt":463,"check":464,"hints":466,"solution":469,"skills":470},"properties-of-numbers.q030","Take out the common factor: **37 × 68 + 37 × 32** = ?",{"kind":215,"answer":465,"tolerance":217},3700,[467,468],"What do the two products have in common?","68 + 32 = 100.","Both products share 37: 37 × (68 + 32) = 37 × 100 = **3,700**. 37 × 68 + 37 × 32 = 3,700.",[23,471],"common factor",{"id":473,"section":23,"level":100,"prompt":474,"check":475,"hints":477,"solution":479,"skills":480},"properties-of-numbers.q031","Take out the common factor: **54 × 83 − 54 × 73** = ?",{"kind":215,"answer":476,"tolerance":217},540,[478],"Distributive over subtraction.","54 × (83 − 73) = 54 × 10 = **540**. 54 × 83 − 54 × 73 = 540.",[23,471],{"id":482,"section":23,"level":100,"prompt":483,"check":484,"hints":495,"solution":497,"skills":498},"properties-of-numbers.q032","Meena wrote 7 × (20 + 3) = 140 + 3 = 143. What is the correct answer and her mistake?",{"kind":48,"options":485,"correct":494},[486,488,490,492],{"id":51,"label":487},"143; no mistake",{"id":54,"label":489},"161; she did not multiply the 3 by 7",{"id":57,"label":491},"161; she should have added first",{"id":60,"label":493},"147; she multiplied 20 by 3",[54],[496],"Does the 7 reach the 3?","The 7 must multiply **both** parts: 7 × 20 + 7 × 3 = 140 + 21 = 161. Check: 7 × 23 = 161.",[23,499],"error analysis",{"id":501,"section":23,"level":100,"prompt":502,"check":503,"hints":506,"solution":508,"skills":509},"properties-of-numbers.q033","An auditorium has 23 rows of 14 seats. Use an area model (20 + 3) × (10 + 4). How many seats?",{"kind":215,"answer":504,"tolerance":217,"unit":505},322,"seats",[507],"Four pieces: tens × tens, tens × ones, ones × tens, ones × ones.","Pieces: 20 × 10 = 200, 20 × 4 = 80, 3 × 10 = 30, 3 × 4 = 12. Total 200 + 80 + 30 + 12 = 322 seats.",[23,510],"area model",{"id":512,"section":23,"level":100,"prompt":513,"check":514,"hints":516,"solution":517,"skills":518},"properties-of-numbers.q034","Work out **8 × 47** by breaking 47 into 40 + 7.",{"kind":215,"answer":515,"tolerance":217},376,[],"8 × 40 + 8 × 7 = 320 + 56 = 376.",[23],{"id":520,"section":23,"level":158,"prompt":521,"check":522,"hints":533,"solution":535,"skills":536},"properties-of-numbers.q035","Which is a correct use of the distributive property?",{"kind":48,"options":523,"correct":532},[524,526,528,530],{"id":51,"label":525},"(60 + 12) ÷ 6 = 10 + 2",{"id":54,"label":527},"60 ÷ (4 + 2) = 15 + 30",{"id":57,"label":529},"2 × (3 × 5) = 6 × 10",{"id":60,"label":531},"3 + (4 × 5) = 7 × 8",[51],[534],"You may split the number being divided, not the divisor.","Division distributes over a sum only on the **left** (the dividend): (60 + 12) ÷ 6 = 10 + 2 = 12. You cannot split the divisor: 60 ÷ 6 = 10, not 15 + 30. Multiplication does not distribute over multiplication, and addition does not distribute over multiplication.",[23,377],{"id":538,"section":23,"level":158,"prompt":539,"check":540,"hints":542,"solution":544,"skills":545},"properties-of-numbers.q036","A shop sells 47 packets at ₹999 each. Total in rupees?",{"kind":215,"answer":541,"tolerance":217,"unit":307},46953,[543],"Multiply by 1,000, then subtract one lot of 47.","999 = 1,000 − 1, so 47 × 999 = 47,000 − 47 = **₹46,953**. 47 × 999 = 46,953.",[23,442],{"id":547,"section":23,"level":158,"prompt":548,"check":549,"hints":551,"solution":553,"skills":554},"properties-of-numbers.q037","Work out **102 × 98** using (a + b)(a − b) = a² − b².",{"kind":215,"answer":550,"tolerance":217},9996,[552],"a = 100, b = 2.","(100 + 2)(100 − 2) = 10,000 − 4 = **9,996**. 102 × 98 = 9,996.",[23,555],"identities",{"id":557,"section":23,"level":408,"prompt":558,"check":559,"hints":561,"solution":563,"skills":564},"properties-of-numbers.q038","Work out **999 × 999** without a calculator.",{"kind":215,"answer":560,"tolerance":217},998001,[562],"Use (a − b)² = a² − 2ab + b².","(1,000 − 1)² = 1,000,000 − 2 × 1,000 + 1 = **998,001**. 999 × 999 = 998,001.",[23,555],{"id":566,"section":27,"level":45,"prompt":567,"check":568,"hints":569,"solution":570,"skills":571},"properties-of-numbers.q039","What is 4,567 × 0?",{"kind":215,"answer":217,"tolerance":217},[],"Any number multiplied by 0 is 0: 4,567 groups of nothing is nothing.",[572],"zero",{"id":574,"section":27,"level":45,"prompt":575,"check":576,"hints":577,"solution":578,"skills":579},"properties-of-numbers.q040","What is 0 ÷ 9?",{"kind":215,"answer":217,"tolerance":217},[],"Sharing 0 among 9 people gives each 0. Check: 0 × 9 = 0. ✓",[572],{"id":581,"section":27,"level":45,"prompt":582,"check":583,"hints":591,"solution":592,"skills":593},"properties-of-numbers.q041","Which number is the **additive identity**?",{"kind":48,"options":584,"correct":590},[585,586,587,589],{"id":51,"label":52},{"id":54,"label":88},{"id":57,"label":588},"10",{"id":60,"label":91},[51],[],"Adding 0 leaves any number unchanged: a + 0 = 0 + a = a. So 0 is the additive identity. (1 is the multiplicative identity.)",[594],"identity",{"id":596,"section":27,"level":45,"prompt":597,"check":598,"hints":609,"solution":610,"skills":611},"properties-of-numbers.q042","Which statement is true?",{"kind":48,"options":599,"correct":608},[600,602,604,606],{"id":51,"label":601},"8 × 1 = 9",{"id":54,"label":603},"8 + 1 = 8",{"id":57,"label":605},"8 × 1 = 8",{"id":60,"label":607},"8 ÷ 1 = 1",[57],[],"Multiplying by 1 changes nothing: 8 × 1 = 8. 8 + 1 = 9 and 8 ÷ 1 = 8.",[594,612],"one",{"id":614,"section":27,"level":45,"prompt":615,"check":616,"hints":618,"solution":619,"skills":620},"properties-of-numbers.q043","What is 58 + 0?",{"kind":215,"answer":617,"tolerance":217},58,[],"Adding 0 changes nothing: 58 + 0 = 58. 0 is the additive identity.",[594],{"id":622,"section":27,"level":100,"prompt":623,"check":624,"hints":635,"solution":637,"skills":638},"properties-of-numbers.q044","Why is 12 ÷ 0 undefined?",{"kind":48,"options":625,"correct":634},[626,628,630,632],{"id":51,"label":627},"Because the answer is 0",{"id":54,"label":629},"Because the answer is 12",{"id":57,"label":631},"Because no number q has q × 0 = 12",{"id":60,"label":633},"Because 12 is even",[57],[636],"Check with multiplication.","Every division can be checked by multiplication: 12 ÷ 3 = 4 because 4 × 3 = 12. For 12 ÷ 0 we would need q × 0 = 12, but q × 0 = 0 for every q. No answer exists, so it is undefined.",[572,639],"division by zero",{"id":641,"section":27,"level":100,"prompt":642,"check":643,"hints":657,"solution":659,"skills":660},"properties-of-numbers.q045","Which of these are equal to 1? (Choose all that apply.)",{"kind":48,"options":644,"correct":656},[645,647,649,651,653],{"id":51,"label":646},"7 ÷ 7",{"id":54,"label":648},"0 ÷ 0",{"id":57,"label":650},"1 × 1 × 1",{"id":60,"label":652},"1 + 0",{"id":654,"label":655},"e","0 × 1",[51,57,60],[658],"Be careful with 0 ÷ 0.","7 ÷ 7 = 1, 1 × 1 × 1 = 1 and 1 + 0 = 1. 0 × 1 = 0. 0 ÷ 0 is **undefined**, not 1, because every number q satisfies q × 0 = 0.",[612,572],{"id":662,"section":27,"level":100,"prompt":663,"check":664,"hints":675,"solution":676,"skills":677},"properties-of-numbers.q046","Why is 0 **not** a true identity for subtraction?",{"kind":48,"options":665,"correct":674},[666,668,670,672],{"id":51,"label":667},"Because 9 − 0 is not 9",{"id":54,"label":669},"Because 0 − 9 is not 9",{"id":57,"label":671},"Because 0 is not a whole number",{"id":60,"label":673},"It is a true identity",[54],[],"An identity must work on both sides. 9 − 0 = 9 works, but 0 − 9 is not 9 (it is not even a whole number). So 0 is only a \"right-hand\" identity for subtraction.",[594],{"id":679,"section":27,"level":100,"prompt":680,"check":681,"hints":682,"solution":684,"skills":685},"properties-of-numbers.q047","If 346 × □ = 0, what number is in the box?",{"kind":215,"answer":217,"tolerance":217},[683],"Zero-product rule.","A product is 0 only if one of the factors is 0. 346 is not 0, so □ = **0**.",[572],{"id":687,"section":27,"level":158,"prompt":688,"check":689,"hints":700,"solution":701,"skills":702},"properties-of-numbers.q048","Why is 0 ÷ 0 left undefined?",{"kind":48,"options":690,"correct":699},[691,693,695,697],{"id":51,"label":692},"Because it equals 0",{"id":54,"label":694},"Because it equals 1",{"id":57,"label":696},"Because every number q satisfies q × 0 = 0, so no single answer can be chosen",{"id":60,"label":698},"Because 0 is not a number",[57],[],"For 0 ÷ 0 = q we need q × 0 = 0. That is true for q = 0, 1, 2, 1,000 … every number. A calculation that could equal anything has no definite value, so mathematicians leave it undefined.",[639,39],{"id":704,"section":27,"level":158,"prompt":705,"check":706,"hints":717,"solution":718,"skills":719},"properties-of-numbers.q049","Riya claims: \"If 1 ÷ 0 = k for some number k, then k × 0 = 1.\" Why does this show 1 ÷ 0 cannot have a value?",{"kind":48,"options":707,"correct":716},[708,710,712,714],{"id":51,"label":709},"Because k would have to be 1",{"id":54,"label":711},"Because k × 0 = 0, so we would get 0 = 1",{"id":57,"label":713},"Because k would be negative",{"id":60,"label":715},"It does not show anything",[54],[],"The zero property says k × 0 = 0 for every k. Together with k × 0 = 1 this gives 0 = 1, which is impossible. So no number k can be 1 ÷ 0. This is a proof by contradiction.",[639,175],{"id":721,"section":27,"level":408,"prompt":722,"check":723,"hints":734,"solution":736,"skills":737},"properties-of-numbers.q050","Find the error. \"Let a = b. Then a² − b² = ab − b², so (a + b)(a − b) = b(a − b). Dividing by (a − b): a + b = b, so 2b = b and 2 = 1.\" Which step is wrong?",{"kind":48,"options":724,"correct":733},[725,727,729,731],{"id":51,"label":726},"a² − b² = ab − b²",{"id":54,"label":728},"Factorising both sides",{"id":57,"label":730},"Dividing by (a − b)",{"id":60,"label":732},"Replacing a by b at the end",[57],[735],"What is a − b when a = b?","Since a = b, a − b = 0. Dividing both sides by (a − b) is dividing by zero, which is not allowed. Every other step is correct.",[639,499],{"id":739,"section":31,"level":45,"prompt":740,"check":741,"hints":748,"solution":750,"skills":751},"properties-of-numbers.q051","Without adding, is 37 + 48 even or odd?",{"kind":48,"options":742,"correct":747},[743,745],{"id":51,"label":744},"Even",{"id":54,"label":746},"Odd",[54],[749],"Look at the last digits: 7 and 8.","odd + even = odd. Check: 37 + 48 = 85, which is odd.",[155],{"id":753,"section":31,"level":45,"prompt":754,"check":755,"hints":760,"solution":761,"skills":762},"properties-of-numbers.q052","Without multiplying, is 23 × 15 even or odd?",{"kind":48,"options":756,"correct":759},[757,758],{"id":51,"label":744},{"id":54,"label":746},[54],[],"odd × odd = odd. Check: 23 × 15 = 345.",[155],{"id":764,"section":31,"level":45,"prompt":765,"check":766,"hints":773,"solution":774,"skills":775},"properties-of-numbers.q053","Is 0 even or odd?",{"kind":48,"options":767,"correct":772},[768,769,770],{"id":51,"label":744},{"id":54,"label":746},{"id":57,"label":771},"Neither",[51],[],"0 is even: 0 things make 0 pairs with nothing left over, and 0 = 2 × 0. It also sits between the odd numbers −1 and 1.",[155,572],{"id":777,"section":31,"level":100,"prompt":778,"check":779,"hints":790,"solution":792,"skills":793},"properties-of-numbers.q054","The sum of **five** odd numbers is always…",{"kind":48,"options":780,"correct":789},[781,783,785,787],{"id":51,"label":782},"even",{"id":54,"label":784},"odd",{"id":57,"label":786},"a multiple of 5",{"id":60,"label":788},"sometimes even, sometimes odd",[54],[791],"Pair the odd numbers two at a time.","Four of them pair up into an even total, and adding the fifth odd number makes it odd. An odd number of odd numbers always has an odd sum.",[155,39],{"id":795,"section":31,"level":100,"prompt":796,"check":797,"hints":799,"solution":801,"skills":802},"properties-of-numbers.q055","How many of these are odd? 45 + 27, 45 × 27, 45 − 27, 45 × 26, 44 + 27",{"kind":215,"answer":798,"tolerance":217},2,[800],"Use the rules: odd ± odd = even; odd × odd = odd.","45 + 27 = 72 (odd + odd, even); 45 × 27 = 1,215 (odd × odd, odd); 45 − 27 = 18 (even); 45 × 26 = 1,170 (even factor, even); 44 + 27 = 71 (odd). So **2** are odd.",[155],{"id":804,"section":31,"level":100,"prompt":805,"check":806,"hints":817,"solution":819,"skills":820},"properties-of-numbers.q056","Can five odd numbers add up to 50?",{"kind":48,"options":807,"correct":816},[808,810,812,814],{"id":51,"label":809},"Yes, for example 10 + 10 + 10 + 10 + 10",{"id":54,"label":811},"Yes, 9 + 9 + 11 + 11 + 11",{"id":57,"label":813},"No, the sum of five odd numbers is always odd",{"id":60,"label":815},"No, because 50 is too big",[57],[818],"Check whether the options really use odd numbers.","10 is even, and 9 + 9 + 11 + 11 + 11 = 51. The sum of five odd numbers is always odd, and 50 is even, so it is impossible.",[155,39],{"id":822,"section":31,"level":158,"prompt":823,"check":824,"hints":833,"solution":834,"skills":835},"properties-of-numbers.q057","The product of two consecutive whole numbers (like 7 × 8) is always…",{"kind":48,"options":825,"correct":832},[826,827,828,830],{"id":51,"label":784},{"id":54,"label":782},{"id":57,"label":829},"a square",{"id":60,"label":831},"a multiple of 3",[54],[],"Of two consecutive numbers, one is even, and anything × even is even. For example 7 × 8 = 56, 12 × 13 = 156.",[155,836],"consecutive numbers",{"id":838,"section":31,"level":158,"prompt":839,"check":840,"hints":851,"solution":853,"skills":854},"properties-of-numbers.q058","Which proof shows odd × odd = odd?",{"kind":48,"options":841,"correct":850},[842,844,846,848],{"id":51,"label":843},"3 × 5 = 15, 7 × 9 = 63, so it is always odd",{"id":54,"label":845},"(2m + 1)(2n + 1) = 2(2mn + m + n) + 1",{"id":57,"label":847},"2m × 2n = 4mn",{"id":60,"label":849},"(2m + 1) + (2n + 1) = 2(m + n + 1)",[54],[852],"A proof must cover every case.","Examples do not prove \"always\". Expanding (2m + 1)(2n + 1) with the distributive law gives 4mn + 2m + 2n + 1 = 2(2mn + m + n) + 1, which is one more than an even number, for every m and n.",[155,175],{"id":856,"section":31,"level":408,"prompt":857,"check":858,"hints":865,"solution":868,"skills":869},"properties-of-numbers.q059","Put a + or − sign in front of each of 1, 2, 3, …, 10. Which total is possible?",{"kind":48,"options":859,"correct":864},[860,861,862,863],{"id":51,"label":52},{"id":54,"label":588},{"id":57,"label":88},{"id":60,"label":384},[57],[866,867],"What is 1 + 2 + … + 10?","Does flipping one sign change the parity?","With all + signs the total is 55, which is odd. Flipping the sign of n changes the total by 2n, an even amount, so the total is always odd. Only 1 is odd among the options, and it can be reached: +1 + 2 + 3 + 4 + 5 + 6 + 7 − 8 − 9 − 10 = 28 − 27 = 1.",[155,870],"invariant",{"id":872,"section":31,"level":408,"prompt":873,"check":874,"hints":875,"solution":877,"skills":878},"properties-of-numbers.q060","Nine cups are upside down. Each move turns over exactly two cups. What is the **smallest** possible number of upside-down cups you can ever reach?",{"kind":215,"answer":5,"tolerance":217},[876],"Does the count stay odd or even?","Each move changes the number of upside-down cups by −2, 0 or +2, so it stays odd. It starts at 9 (odd), so it can never be 0. It can reach 1 (turn two upside-down cups each move: 9 → 7 → 5 → 3 → 1). The smallest is **1**.",[155,870],{"id":880,"section":35,"level":45,"prompt":881,"check":882,"hints":884,"solution":885,"skills":886},"properties-of-numbers.q061","Work out **16 × 5** by halving and doubling (8 × 10).",{"kind":215,"answer":883,"tolerance":217},80,[],"Halve 16 to 8, double 5 to 10: 8 × 10 = 80.",[887],"doubling and halving",{"id":889,"section":35,"level":45,"prompt":890,"check":891,"hints":893,"solution":894,"skills":895},"properties-of-numbers.q062","Compensation: **198 + 67** = ? (Think 200 + 65.)",{"kind":215,"answer":892,"tolerance":217},265,[],"Move 2 from 67 to 198: 200 + 65 = 265.",[896],"compensation",{"id":898,"section":35,"level":100,"prompt":899,"check":900,"hints":902,"solution":904,"skills":905},"properties-of-numbers.q063","Work out **36 × 25** using × 100 ÷ 4.",{"kind":215,"answer":901,"tolerance":217},900,[903],"25 = 100 ÷ 4.","36 × 25 = 36 × 100 ÷ 4 = 3,600 ÷ 4 = **900**. 36 × 25 = 900.",[906],"times 25",{"id":908,"section":35,"level":100,"prompt":909,"check":910,"hints":912,"solution":914,"skills":915},"properties-of-numbers.q064","Work out **64 × 101**.",{"kind":215,"answer":911,"tolerance":217},6464,[913],"101 = 100 + 1.","64 × 100 + 64 = 6,400 + 64 = **6,464**. 64 × 101 = 6,464.",[916],"times 101",{"id":918,"section":35,"level":100,"prompt":919,"check":920,"hints":922,"solution":925,"skills":926},"properties-of-numbers.q065","Work out **73 × 11** using the digit-sum trick.",{"kind":215,"answer":921,"tolerance":217},803,[923,924],"Put the sum of the digits in the middle.","Carry if the sum is 10 or more.","Digits 7 and 3, middle 7 + 3 = 10: write 0 and carry 1 to the 7, giving **803**. 73 × 11 = 730 + 73 = 803.",[927],"times 11",{"id":929,"section":35,"level":100,"prompt":930,"check":931,"hints":933,"solution":934,"skills":935},"properties-of-numbers.q066","Compensation in subtraction: **83 − 38** = ? (Add 2 to both numbers.)",{"kind":215,"answer":932,"tolerance":217},45,[],"83 − 38 = 85 − 40 = **45**. Adding the same amount to both numbers keeps the gap the same. 83 − 38 = 45.",[896],{"id":937,"section":35,"level":100,"prompt":938,"check":939,"hints":942,"solution":943,"skills":944},"properties-of-numbers.q067","A cricket team scores 6 runs off each of 29 balls. How many runs? (6 × 30 − 6.)",{"kind":215,"answer":940,"tolerance":217,"unit":941},174,"runs",[],"6 × 30 − 6 = 180 − 6 = 174 runs.",[23,405],{"id":946,"section":35,"level":158,"prompt":947,"check":948,"hints":950,"solution":952,"skills":953},"properties-of-numbers.q068","Work out **125 × 48** using 125 × 8 = 1,000.",{"kind":215,"answer":949,"tolerance":217},6000,[951],"Split 48 into 8 × 6.","48 = 8 × 6, so 125 × 48 = (125 × 8) × 6 = 1,000 × 6 = **6,000**. 125 × 48 = 6,000.",[19,954],"times 125",{"id":956,"section":35,"level":158,"prompt":957,"check":958,"hints":960,"solution":961,"skills":962},"properties-of-numbers.q069","Work out **35 × 18** by doubling and halving.",{"kind":215,"answer":959,"tolerance":217},630,[],"Double 35 to 70, halve 18 to 9: 70 × 9 = 630.",[887],{"id":964,"section":35,"level":158,"prompt":965,"check":966,"hints":968,"solution":970,"skills":971},"properties-of-numbers.q070","A kirana bill: ₹245, ₹90, ₹155, ₹110, ₹45, ₹55. Pair them to find the total.",{"kind":215,"answer":967,"tolerance":217,"unit":307},700,[969],"Find pairs that make round hundreds.","(245 + 155) + (90 + 110) + (45 + 55) = 400 + 200 + 100 = **₹700**. 245 + 90 + 155 + 110 + 45 + 55 = 700.",[19,442],{"id":973,"section":35,"level":158,"prompt":974,"check":975,"hints":977,"solution":979,"skills":980},"properties-of-numbers.q071","Work out **65 × 65** using the \"ends in 5\" trick.",{"kind":215,"answer":976,"tolerance":217},4225,[978],"Multiply the tens digit by the next number up.","6 × 7 = 42, then write 25: **4,225**. 65 × 65 = 4,225.",[981],"squares",{"id":983,"section":35,"level":408,"prompt":984,"check":985,"hints":987,"solution":989,"skills":990},"properties-of-numbers.q072","Find **1 + 2 + 3 + … + 60** using pairs.",{"kind":215,"answer":986,"tolerance":217},1830,[988],"How many pairs of 61 are there?","Pair first with last: 1 + 60 = 61, 2 + 59 = 61, … There are 30 pairs, so 30 × 61 = 1,830.",[15,19],{"id":992,"section":35,"level":408,"prompt":993,"check":994,"hints":996,"solution":998,"skills":999},"properties-of-numbers.q073","Work out **4 × 125 × 25 × 8** in your head.",{"kind":215,"answer":995,"tolerance":217},100000,[997],"Pair 4 with 25 and 125 with 8.","Regroup: (4 × 25) × (125 × 8) = 100 × 1,000 = **100,000**. 4 × 125 × 25 × 8 = 100,000.",[19,15],{"id":1001,"section":39,"level":45,"prompt":1002,"check":1003,"hints":1014,"solution":1015,"skills":1016},"properties-of-numbers.q074","Which is a **counterexample** to \"subtraction is commutative\"?",{"kind":48,"options":1004,"correct":1013},[1005,1007,1009,1011],{"id":51,"label":1006},"7 − 7 = 7 − 7",{"id":54,"label":1008},"9 − 4 = 5 but 4 − 9 is not 5",{"id":57,"label":1010},"9 + 4 = 4 + 9",{"id":60,"label":1012},"10 − 0 = 10",[54],[],"A counterexample is one case where the claim fails. 9 − 4 = 5, but 4 − 9 is not 5, so subtraction is not commutative.",[117],{"id":1018,"section":39,"level":100,"prompt":1019,"check":1020,"hints":1029,"solution":1031,"skills":1032},"properties-of-numbers.q075","\"When you multiply two whole numbers, the answer is bigger than both.\" This is…",{"kind":48,"options":1021,"correct":1028},[1022,1024,1026],{"id":51,"label":1023},"always true",{"id":54,"label":1025},"sometimes true",{"id":57,"label":1027},"never true",[54],[1030],"Try 0 and 1.","True for 3 × 4 = 12. False for 3 × 1 = 3 and 3 × 0 = 0. It is true exactly when both numbers are at least 2.",[1033],"always sometimes never",{"id":1035,"section":39,"level":100,"prompt":1036,"check":1037,"hints":1043,"solution":1045,"skills":1046},"properties-of-numbers.q076","\"The sum of three consecutive whole numbers is a multiple of 3.\" This is…",{"kind":48,"options":1038,"correct":1042},[1039,1040,1041],{"id":51,"label":1023},{"id":54,"label":1025},{"id":57,"label":1027},[51],[1044],"Write the numbers around the middle one.","The numbers are (m − 1), m, (m + 1) for a middle number m, so the sum is 3m. For example 12 + 13 + 14 = 39 = 3 × 13.",[1033,175],{"id":1048,"section":39,"level":100,"prompt":1049,"check":1050,"hints":1056,"solution":1057,"skills":1058},"properties-of-numbers.q077","\"The sum of two consecutive whole numbers is even.\" This is…",{"kind":48,"options":1051,"correct":1055},[1052,1053,1054],{"id":51,"label":1023},{"id":54,"label":1025},{"id":57,"label":1027},[57],[],"One of two consecutive numbers is even and the other odd, so the sum is always odd. The claim is never true.",[1033,155],{"id":1060,"section":39,"level":100,"prompt":1061,"check":1062,"hints":1073,"solution":1074,"skills":1075},"properties-of-numbers.q078","Anu tested \"a × b = b × a\" with 20 pairs and it always worked. Which is the best conclusion?",{"kind":48,"options":1063,"correct":1072},[1064,1066,1068,1070],{"id":51,"label":1065},"It is proved for all numbers",{"id":54,"label":1067},"It is probably true; a reason such as rotating an array proves it",{"id":57,"label":1069},"It is false",{"id":60,"label":1071},"She needs 100 more examples",[54],[],"Examples make us suspect a rule but cannot prove it for infinitely many numbers. The array argument (a rows of b dots, turned round, is b rows of a dots) proves it for every pair.",[175,39],{"id":1077,"section":39,"level":158,"prompt":1078,"check":1079,"hints":1090,"solution":1092,"skills":1093},"properties-of-numbers.q079","Which operation on whole numbers is commutative but **not** associative?",{"kind":48,"options":1080,"correct":1089},[1081,1083,1085,1087],{"id":51,"label":1082},"a + b",{"id":54,"label":1084},"The average (a + b) ÷ 2",{"id":57,"label":1086},"a − b",{"id":60,"label":1088},"a × b",[54],[1091],"Test with 0, 4 and 8.","avg(a, b) = avg(b, a), so it is commutative. But avg(avg(0, 4), 8) = avg(2, 8) = 5, while avg(0, avg(4, 8)) = avg(0, 6) = 3. Not associative.",[1094,117],"operations",{"id":1096,"section":39,"level":158,"prompt":1097,"check":1098,"hints":1109,"solution":1111,"skills":1112},"properties-of-numbers.q080","Is \"a + (b × c) = (a + b) × (a + c)\" true for all whole numbers?",{"kind":48,"options":1099,"correct":1108},[1100,1102,1104,1106],{"id":51,"label":1101},"Yes, it is the distributive law",{"id":54,"label":1103},"No: a = 1, b = 2, c = 3 gives 7 and 12",{"id":57,"label":1105},"Yes, but only for even numbers",{"id":60,"label":1107},"No, it is never true",[54],[1110],"Try a = 1, b = 2, c = 3.","Multiplication distributes over addition, but addition does not distribute over multiplication. 1 + 2 × 3 = 7 but (1 + 2)(1 + 3) = 12. (It happens to work when a = 0, so \"never true\" is also wrong.)",[23,117],{"id":1114,"section":39,"level":158,"prompt":1115,"check":1116,"hints":1127,"solution":1129,"skills":1130},"properties-of-numbers.q081","A square number can never leave remainder 2 or 3 when divided by 4. Which of these could be a perfect square?",{"kind":48,"options":1117,"correct":1126},[1118,1120,1122,1124],{"id":51,"label":1119},"2,738",{"id":54,"label":1121},"5,183",{"id":57,"label":1123},"6,241",{"id":60,"label":1125},"9,002",[57],[1128],"Only the last two digits matter for division by 4.","Look at the last two digits and divide by 4: 38 leaves 2, 83 leaves 3, 41 leaves 1, 02 leaves 2. Only 6,241 passes, and indeed 79 × 79 = 6,241.",[981,39],{"id":1132,"section":39,"level":158,"prompt":1133,"check":1134,"hints":1136,"solution":1138,"skills":1139},"properties-of-numbers.q082","How many rectangular rows × columns arrangements are there for 36 chairs, counting 4 × 9 and 9 × 4 as different?",{"kind":215,"answer":1135,"tolerance":217},9,[1137],"List the factors of 36.","Factor pairs of 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × 6. The first four each give two arrangements (turned round), and 6 × 6 gives one: 4 × 2 + 1 = **9**.",[15,1140],"factors",{"id":1142,"section":39,"level":408,"prompt":1143,"check":1144,"hints":1155,"solution":1157,"skills":1158},"properties-of-numbers.q083","Every six-digit number of the form abcabc (like 725,725) is divisible by 13. Why?",{"kind":48,"options":1145,"correct":1154},[1146,1148,1150,1152],{"id":51,"label":1147},"Because its digits add to a multiple of 13",{"id":54,"label":1149},"Because abcabc = abc × 1,001 and 1,001 = 7 × 11 × 13",{"id":57,"label":1151},"Because it is even",{"id":60,"label":1153},"It is only true for some such numbers",[54],[1156],"Write abcabc as abc × 1,000 + abc.","abcabc = abc × 1,000 + abc = abc × 1,001 (distributive law), and 1,001 = 7 × 11 × 13. So it is divisible by 7, 11 and 13. For example 725,725 = 725 × 1,001.",[23,175],{"id":1160,"section":39,"level":408,"prompt":1161,"check":1162,"hints":1173,"solution":1175,"skills":1176},"properties-of-numbers.q084","The operation a ⊲ b = a (\"keep the first number\") is…",{"kind":48,"options":1163,"correct":1172},[1164,1166,1168,1170],{"id":51,"label":1165},"commutative and associative",{"id":54,"label":1167},"commutative only",{"id":57,"label":1169},"associative only",{"id":60,"label":1171},"neither",[57],[1174],"Test order with 3 and 5.","(a ⊲ b) ⊲ c = a and a ⊲ (b ⊲ c) = a, so it is associative. But 3 ⊲ 5 = 3 while 5 ⊲ 3 = 5, so it is not commutative. The two properties are independent.",[1094,39],{"id":1178,"section":39,"level":408,"prompt":1179,"check":1180,"hints":1181,"solution":1183,"skills":1184},"properties-of-numbers.q085","What is the last digit of 3 to the power 50 (3 multiplied by itself 50 times)?",{"kind":215,"answer":1135,"tolerance":217},[1182],"Write down the last digits of 3, 9, 27, 81, 243, …","Last digits of powers of 3 cycle: 3, 9, 7, 1, 3, 9, … (cycle of 4). 50 = 12 × 4 + 2, so 3⁵⁰ ends like 3², that is **9**.",[1185,19],"patterns",[1187,1188,1189,1190,1191,1192,1193,1194,1195,1196],"properties-of-numbers-ncert-class6-whole-numbers","properties-of-numbers-ncert-class7-integers","properties-of-numbers-ncert-class8-rational-numbers","properties-of-numbers-mathsisfun-properties","properties-of-numbers-mathsisfun-divide-by-zero","properties-of-numbers-wiki-commutative","properties-of-numbers-wiki-distributive","properties-of-numbers-wiki-division-by-zero","properties-of-numbers-wiki-parity","properties-of-numbers-wiki-brahmagupta","needs_review",{"generatedBy":1199,"notes":1200},"claude-code","Generated with a Python script; every numeric answer computed and asserted. Pending owner review.","2fc81768e0be1fdd9c7e9fad42d185eb5ffbadda3bcb03a8d0e69429de8760ab",{"logic:questions":1203,"source:properties-of-numbers-mathsisfun-divide-by-zero":1204,"source:properties-of-numbers-mathsisfun-properties":1205,"source:properties-of-numbers-ncert-class6-whole-numbers":1206,"source:properties-of-numbers-ncert-class7-integers":1207,"source:properties-of-numbers-ncert-class8-rational-numbers":1208,"source:properties-of-numbers-wiki-brahmagupta":1209,"source:properties-of-numbers-wiki-commutative":1210,"source:properties-of-numbers-wiki-distributive":1211,"source:properties-of-numbers-wiki-division-by-zero":1212,"source:properties-of-numbers-wiki-parity":1213},"e7fd7c240a65bea1bff6277f7af7cca65e2c7fcb13c6d756cb943d53f3cbc948","f8a42fd8c82c266f4310450d74f1ae0449b3dba177afcf000b24aafa2839b629","a37b85a36742e06f34cdc083710eacdd662e2f7493395766e4b8c5632bfade3a","56b97dbf2988a58e1aa10f693ec9dd1039e1de14e795b1bdfa19980b6b18c1fa","aab8fe53a32660ef6307b4dbd6f5af02319957d9aecf3c0396895bc01ab46e2e","acb8b92cad8387724d72528e70ef68195351e5b2f03b580c5da191fde82dc67a","32e7279091a5877afe54d6da75cdd06a04f2ff8677338777578592d6dd98a338","a3e0fb47a2115f5962c9795b138bf38cd9c2c4874bc66e1b834717b9a01908c1","befe26bd7ec4e1c3911549d33a91844ae3eb5bd63388711f61822d398d3d576f","046256cc76cdb1b7cca72a1b391bef686855afdfb6a68d18b45a38723d1fe93d","01c5bcf03886aec6f9c2c18c9ec5333b19ce1d2ca6bfb4f8544c4ca9f1578e27","preview-7e1cbbcc4f",1789899599959]