Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 648 · 40 questions

How to solve it (Patterns repeat): Divisibility: 3 and 9 — digit sum; 4 — last two digits; 8 — last three digits; 6 — even and divisible by 3; 11 — alternate digit sums differ by 0 or a multiple of 11. Remainder: find the biggest multiple not more than the number and take it away; to make a number divisible, add divisor − remainder. Units digit of a power: write the cycle (7, 9, 3, 1 for 7), divide the power by the cycle length and use the remainder.

1. 1.Which of these numbers is divisible by 9? (a) 6449 (b) 6457 (c) 6453 (d) 6466
2. 2.Which of these numbers is divisible by 4? (a) 2407 (b) 2411 (c) 2415 (d) 2408
3. 3.Which of these numbers is divisible by 3? (a) 2372 (b) 2365 (c) 2369 (d) 2367
4. 4.Which of these numbers is divisible by 9? (a) 4962 (b) 4981 (c) 4968 (d) 4971
5. 5.Which of these numbers is divisible by 11? (a) 3205 (b) 3198 (c) 3215 (d) 3201
6. 6.What is the smallest number you must add to 239 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 1
7. 7.What is the remainder when 161 is divided by 10? (a) 0 (b) 2 (c) 9 (d) 1
8. 8.What is the remainder when 131 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
9. 9.Which of these numbers is divisible by 9? (a) 2570 (b) 2574 (c) 2584 (d) 2582
10. 10.What is the remainder when 401 is divided by 10? (a) 0 (b) 9 (c) 1 (d) 2
11. 11.What is the smallest number you must add to 234 to make it divisible by 10? (a) 6 (b) 5 (c) 7 (d) 4
12. 12.What is the remainder when 193 is divided by 3? (a) 1 (b) 0 (c) 2
13. 13.What is the remainder when 59 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
14. 14.Which of these numbers is divisible by 8? (a) 1520 (b) 1529 (c) 1514 (d) 1525
15. 15.What is the remainder when 523 is divided by 9? (a) 8 (b) 2 (c) 0 (d) 1
16. 16.What is the smallest number you must add to 433 to make it divisible by 11? (a) 6 (b) 7 (c) 8 (d) 4
17. 17.What is the smallest number you must add to 461 to make it divisible by 8? (a) 2 (b) 4 (c) 5 (d) 3
18. 18.Which of these numbers is divisible by 8? (a) 4344 (b) 4345 (c) 4342 (d) 4359
19. 19.What is the smallest number you must add to 849 to make it divisible by 11? (a) 2 (b) 10 (c) 9 (d) 8
20. 20.Which of these numbers is divisible by 6? (a) 499 (b) 493 (c) 488 (d) 492
21. 21.Which of these numbers is divisible by 3? (a) 1207 (b) 1204 (c) 1211 (d) 1206
22. 22.What is the smallest number you must add to 373 to make it divisible by 10? (a) 3 (b) 7 (c) 8 (d) 6
23. 23.What is the remainder when 461 is divided by 10? (a) 9 (b) 0 (c) 1 (d) 2
24. 24.What is the remainder when 269 is divided by 11? (a) 6 (b) 4 (c) 7 (d) 5
25. 25.What is the remainder when 295 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
26. 26.Which of these numbers is divisible by 3? (a) 205 (b) 208 (c) 204 (d) 202
27. 27.What is the remainder when 147 is divided by 12? (a) 9 (b) 2 (c) 4 (d) 3
28. 28.What is the units digit of 9^46? (a) 2 (b) 1 (c) 9 (d) 0
29. 29.What is the units digit of 4^27? (a) 3 (b) 6 (c) 5 (d) 4
30. 30.What is the smallest number you must add to 444 to make it divisible by 10? (a) 5 (b) 4 (c) 6 (d) 7
31. 31.What is the units digit of 7^56? (a) 1 (b) 3 (c) 7 (d) 9
32. 32.Which of these numbers is divisible by 9? (a) 1503 (b) 1505 (c) 1515 (d) 1502
33. 33.Which of these numbers is divisible by 9? (a) 3393 (b) 3406 (c) 3386 (d) 3400
34. 34.Which of these numbers is divisible by 4? (a) 1450 (b) 1444 (c) 1447 (d) 1442
35. 35.Which of these numbers is divisible by 6? (a) 2072 (b) 2063 (c) 2064 (d) 2067
36. 36.Which of these numbers is divisible by 9? (a) 5721 (b) 5715 (c) 5714 (d) 5727
37. 37.Which of these numbers is divisible by 11? (a) 9044 (b) 9027 (c) 9031 (d) 9037
38. 38.What is the remainder when 67 is divided by 6? (a) 2 (b) 1 (c) 5 (d) 0
39. 39.Which of these numbers is divisible by 11? (a) 605 (b) 617 (c) 611 (d) 602
40. 40.What is the remainder when 584 is divided by 10? (a) 5 (b) 3 (c) 6 (d) 4

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad