Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 328 · 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 smallest number you must add to 181 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
2. 2.Which of these numbers is divisible by 3? (a) 494 (b) 499 (c) 497 (d) 495
3. 3.What is the remainder when 244 is divided by 7? (a) 1 (b) 5 (c) 6 (d) 4
4. 4.What is the smallest number you must add to 96 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
5. 5.Which of these numbers is divisible by 8? (a) 4216 (b) 4228 (c) 4218 (d) 4211
6. 6.What is the units digit of 2^57? (a) 8 (b) 4 (c) 2 (d) 6
7. 7.What is the units digit of 8^20? (a) 2 (b) 6 (c) 4 (d) 8
8. 8.What is the remainder when 317 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
9. 9.What is the remainder when 193 is divided by 6? (a) 1 (b) 5 (c) 0 (d) 2
10. 10.What is the smallest number you must add to 113 to make it divisible by 10? (a) 8 (b) 3 (c) 6 (d) 7
11. 11.What is the smallest number you must add to 148 to make it divisible by 7? (a) 5 (b) 1 (c) 6 (d) 4
12. 12.What is the units digit of 7^34? (a) 7 (b) 3 (c) 9 (d) 1
13. 13.What is the units digit of 2^52? (a) 2 (b) 4 (c) 6 (d) 8
14. 14.What is the remainder when 94 is divided by 6? (a) 4 (b) 2 (c) 5 (d) 3
15. 15.What is the units digit of 8^45? (a) 6 (b) 8 (c) 4 (d) 2
16. 16.Which of these numbers is divisible by 11? (a) 2156 (b) 2148 (c) 2175 (d) 2159
17. 17.What is the units digit of 3^54? (a) 7 (b) 3 (c) 1 (d) 9
18. 18.What is the units digit of 9^60? (a) 9 (b) 2 (c) 0 (d) 1
19. 19.What is the smallest number you must add to 277 to make it divisible by 10? (a) 7 (b) 4 (c) 2 (d) 3
20. 20.What is the smallest number you must add to 793 to make it divisible by 12? (a) 9 (b) 11 (c) 10 (d) 1
21. 21.What is the units digit of 3^60? (a) 7 (b) 9 (c) 3 (d) 1
22. 22.What is the units digit of 7^14? (a) 9 (b) 3 (c) 7 (d) 1
23. 23.What is the smallest number you must add to 239 to make it divisible by 4? (a) 3 (b) 1 (c) 2
24. 24.What is the units digit of 2^56? (a) 6 (b) 2 (c) 4 (d) 8
25. 25.What is the units digit of 9^11? (a) 1 (b) 9 (c) 8 (d) 7
26. 26.Which of these numbers is divisible by 11? (a) 1437 (b) 1453 (c) 1444 (d) 1441
27. 27.Which of these numbers is divisible by 4? (a) 750 (b) 741 (c) 744 (d) 745
28. 28.Which of these numbers is divisible by 11? (a) 2715 (b) 2686 (c) 2695 (d) 2698
29. 29.What is the units digit of 3^41? (a) 7 (b) 1 (c) 9 (d) 3
30. 30.What is the smallest number you must add to 225 to make it divisible by 10? (a) 4 (b) 6 (c) 7 (d) 5
31. 31.Which of these numbers is divisible by 9? (a) 1476 (b) 1471 (c) 1477 (d) 1488
32. 32.What is the remainder when 321 is divided by 8? (a) 1 (b) 2 (c) 0 (d) 7
33. 33.Which of these numbers is divisible by 11? (a) 4867 (b) 4854 (c) 4851 (d) 4843
34. 34.What is the remainder when 376 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 5
35. 35.Which of these numbers is divisible by 8? (a) 3522 (b) 3516 (c) 3512 (d) 3509
36. 36.Which of these numbers is divisible by 9? (a) 6552 (b) 6566 (c) 6553 (d) 6550
37. 37.What is the smallest number you must add to 187 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
38. 38.What is the units digit of 9^30? (a) 0 (b) 2 (c) 9 (d) 1
39. 39.Which of these numbers is divisible by 11? (a) 4994 (b) 4988 (c) 4997 (d) 5010
40. 40.What is the smallest number you must add to 632 to make it divisible by 12? (a) 4 (b) 3 (c) 8 (d) 5

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