Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 298 · 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.What is the remainder when 431 is divided by 6? (a) 5 (b) 1 (c) 4 (d) 3
2. 2.What is the remainder when 274 is divided by 7? (a) 1 (b) 2 (c) 6 (d) 0
3. 3.What is the units digit of 7^57? (a) 3 (b) 9 (c) 7 (d) 1
4. 4.What is the units digit of 3^11? (a) 9 (b) 1 (c) 3 (d) 7
5. 5.What is the remainder when 298 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
6. 6.Which of these numbers is divisible by 3? (a) 636 (b) 638 (c) 634 (d) 641
7. 7.What is the remainder when 325 is divided by 9? (a) 8 (b) 2 (c) 1 (d) 0
8. 8.What is the remainder when 155 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
9. 9.What is the smallest number you must add to 89 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
10. 10.What is the smallest number you must add to 34 to make it divisible by 3? (a) 1 (b) 2
11. 11.What is the remainder when 202 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
12. 12.What is the units digit of 3^45? (a) 7 (b) 9 (c) 3 (d) 1
13. 13.What is the units digit of 7^58? (a) 1 (b) 3 (c) 7 (d) 9
14. 14.Which of these numbers is divisible by 9? (a) 1375 (b) 1351 (c) 1363 (d) 1359
15. 15.Which of these numbers is divisible by 11? (a) 9632 (b) 9642 (c) 9636 (d) 9657
16. 16.What is the smallest number you must add to 623 to make it divisible by 8? (a) 2 (b) 7 (c) 1 (d) 3
17. 17.What is the remainder when 810 is divided by 11? (a) 4 (b) 8 (c) 7 (d) 6
18. 18.Which of these numbers is divisible by 4? (a) 3470 (b) 3464 (c) 3463 (d) 3467
19. 19.Which of these numbers is divisible by 6? (a) 3690 (b) 3695 (c) 3689 (d) 3698
20. 20.What is the units digit of 9^37? (a) 1 (b) 7 (c) 9 (d) 8
21. 21.Which of these numbers is divisible by 6? (a) 4184 (b) 4193 (c) 4181 (d) 4182
22. 22.What is the units digit of 9^13? (a) 9 (b) 7 (c) 8 (d) 1
23. 23.Which of these numbers is divisible by 11? (a) 2925 (b) 2932 (c) 2909 (d) 2915
24. 24.What is the smallest number you must add to 79 to make it divisible by 4? (a) 2 (b) 3 (c) 1
25. 25.Which of these numbers is divisible by 9? (a) 3582 (b) 3595 (c) 3577 (d) 3588
26. 26.What is the units digit of 4^49? (a) 3 (b) 5 (c) 4 (d) 6
27. 27.Which of these numbers is divisible by 8? (a) 4269 (b) 4277 (c) 4264 (d) 4257
28. 28.What is the remainder when 145 is divided by 9? (a) 2 (b) 8 (c) 1 (d) 0
29. 29.What is the smallest number you must add to 749 to make it divisible by 11? (a) 8 (b) 10 (c) 1 (d) 9
30. 30.What is the remainder when 172 is divided by 12? (a) 5 (b) 4 (c) 3 (d) 8
31. 31.Which of these numbers is divisible by 4? (a) 3190 (b) 3187 (c) 3183 (d) 3184
32. 32.Which of these numbers is divisible by 8? (a) 5176 (b) 5177 (c) 5186 (d) 5173
33. 33.What is the units digit of 8^26? (a) 8 (b) 6 (c) 4 (d) 2
34. 34.What is the remainder when 598 is divided by 9? (a) 4 (b) 6 (c) 3 (d) 5
35. 35.What is the remainder when 145 is divided by 3? (a) 0 (b) 2 (c) 1
36. 36.What is the smallest number you must add to 153 to make it divisible by 12? (a) 4 (b) 3 (c) 2 (d) 9
37. 37.What is the remainder when 192 is divided by 11? (a) 7 (b) 6 (c) 5 (d) 4
38. 38.What is the smallest number you must add to 194 to make it divisible by 12? (a) 11 (b) 2 (c) 9 (d) 10
39. 39.What is the units digit of 2^36? (a) 4 (b) 6 (c) 8 (d) 2
40. 40.What is the smallest number you must add to 367 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3

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