Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 712 · 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) 3780 (b) 3775 (c) 3781 (d) 3794
2. 2.What is the remainder when 81 is divided by 5? (a) 0 (b) 4 (c) 1 (d) 2
3. 3.What is the smallest number you must add to 239 to make it divisible by 9? (a) 6 (b) 3 (c) 5 (d) 4
4. 4.What is the units digit of 2^46? (a) 2 (b) 8 (c) 4 (d) 6
5. 5.Which of these numbers is divisible by 4? (a) 3002 (b) 3007 (c) 2998 (d) 3000
6. 6.What is the smallest number you must add to 121 to make it divisible by 10? (a) 1 (b) 8 (c) 9 (d) 7
7. 7.Which of these numbers is divisible by 9? (a) 3545 (b) 3522 (c) 3532 (d) 3528
8. 8.What is the smallest number you must add to 96 to make it divisible by 7? (a) 2 (b) 3 (c) 5 (d) 1
9. 9.Which of these numbers is divisible by 9? (a) 7452 (b) 7457 (c) 7462 (d) 7446
10. 10.What is the smallest number you must add to 409 to make it divisible by 9? (a) 4 (b) 5 (c) 6 (d) 7
11. 11.What is the remainder when 210 is divided by 8? (a) 6 (b) 2 (c) 1 (d) 3
12. 12.What is the units digit of 7^59? (a) 7 (b) 1 (c) 9 (d) 3
13. 13.What is the units digit of 8^50? (a) 8 (b) 2 (c) 6 (d) 4
14. 14.What is the units digit of 7^36? (a) 3 (b) 9 (c) 7 (d) 1
15. 15.What is the remainder when 468 is divided by 7? (a) 6 (b) 5 (c) 4 (d) 1
16. 16.What is the remainder when 111 is divided by 5? (a) 0 (b) 4 (c) 2 (d) 1
17. 17.Which of these numbers is divisible by 11? (a) 4967 (b) 4951 (c) 4961 (d) 4980
18. 18.Which of these numbers is divisible by 8? (a) 2731 (b) 2726 (c) 2728 (d) 2737
19. 19.What is the remainder when 229 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
20. 20.What is the remainder when 239 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
21. 21.What is the remainder when 733 is divided by 10? (a) 4 (b) 3 (c) 2 (d) 7
22. 22.Which of these numbers is divisible by 11? (a) 9058 (b) 9053 (c) 9073 (d) 9052
23. 23.What is the smallest number you must add to 355 to make it divisible by 9? (a) 6 (b) 4 (c) 7 (d) 5
24. 24.What is the remainder when 138 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
25. 25.Which of these numbers is divisible by 4? (a) 587 (b) 584 (c) 590 (d) 583
26. 26.Which of these numbers is divisible by 11? (a) 2486 (b) 2478 (c) 2500 (d) 2493
27. 27.What is the remainder when 674 is divided by 12? (a) 10 (b) 1 (c) 2 (d) 3
28. 28.What is the smallest number you must add to 529 to make it divisible by 9? (a) 2 (b) 3 (c) 1 (d) 7
29. 29.What is the units digit of 8^14? (a) 6 (b) 8 (c) 2 (d) 4
30. 30.What is the remainder when 107 is divided by 8? (a) 4 (b) 3 (c) 2 (d) 5
31. 31.Which of these numbers is divisible by 9? (a) 5420 (b) 5427 (c) 5437 (d) 5428
32. 32.What is the units digit of 4^34? (a) 4 (b) 6 (c) 5 (d) 7
33. 33.Which of these numbers is divisible by 3? (a) 2207 (b) 2208 (c) 2210 (d) 2212
34. 34.Which of these numbers is divisible by 8? (a) 1134 (b) 1128 (c) 1138 (d) 1126
35. 35.What is the smallest number you must add to 49 to make it divisible by 4? (a) 1 (b) 3 (c) 2
36. 36.What is the remainder when 155 is divided by 3? (a) 1 (b) 2 (c) 0
37. 37.What is the remainder when 73 is divided by 3? (a) 2 (b) 0 (c) 1
38. 38.What is the remainder when 94 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
39. 39.Which of these numbers is divisible by 11? (a) 9121 (b) 9109 (c) 9119 (d) 9139
40. 40.What is the remainder when 296 is divided by 5? (a) 4 (b) 1 (c) 0 (d) 2

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