Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 759 · 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 257 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
2. 2.What is the units digit of 7^14? (a) 7 (b) 1 (c) 9 (d) 3
3. 3.What is the remainder when 561 is divided by 8? (a) 7 (b) 2 (c) 1 (d) 0
4. 4.Which of these numbers is divisible by 6? (a) 2526 (b) 2530 (c) 2523 (d) 2535
5. 5.What is the units digit of 4^17? (a) 5 (b) 3 (c) 6 (d) 4
6. 6.What is the smallest number you must add to 93 to make it divisible by 9? (a) 3 (b) 7 (c) 5 (d) 6
7. 7.Which of these numbers is divisible by 3? (a) 2386 (b) 2389 (c) 2384 (d) 2385
8. 8.What is the smallest number you must add to 181 to make it divisible by 3? (a) 2 (b) 1
9. 9.Which of these numbers is divisible by 3? (a) 2227 (b) 2223 (c) 2222 (d) 2224
10. 10.What is the smallest number you must add to 417 to make it divisible by 11? (a) 10 (b) 2 (c) 3 (d) 1
11. 11.What is the smallest number you must add to 310 to make it divisible by 12? (a) 2 (b) 10 (c) 3 (d) 1
12. 12.What is the smallest number you must add to 189 to make it divisible by 11? (a) 2 (b) 9 (c) 8 (d) 10
13. 13.What is the smallest number you must add to 150 to make it divisible by 4? (a) 2 (b) 3 (c) 1
14. 14.What is the units digit of 3^27? (a) 3 (b) 1 (c) 7 (d) 9
15. 15.What is the units digit of 8^60? (a) 8 (b) 6 (c) 2 (d) 4
16. 16.What is the remainder when 797 is divided by 10? (a) 3 (b) 7 (c) 6 (d) 8
17. 17.What is the units digit of 8^48? (a) 2 (b) 8 (c) 6 (d) 4
18. 18.What is the smallest number you must add to 123 to make it divisible by 8? (a) 6 (b) 5 (c) 4 (d) 3
19. 19.Which of these numbers is divisible by 3? (a) 523 (b) 517 (c) 520 (d) 519
20. 20.What is the remainder when 426 is divided by 10? (a) 6 (b) 7 (c) 5 (d) 4
21. 21.Which of these numbers is divisible by 9? (a) 4185 (b) 4189 (c) 4195 (d) 4177
22. 22.Which of these numbers is divisible by 6? (a) 2730 (b) 2738 (c) 2729 (d) 2735
23. 23.What is the units digit of 9^52? (a) 9 (b) 1 (c) 0 (d) 2
24. 24.Which of these numbers is divisible by 9? (a) 5900 (b) 5891 (c) 5895 (d) 5905
25. 25.What is the smallest number you must add to 377 to make it divisible by 12? (a) 5 (b) 8 (c) 7 (d) 6
26. 26.What is the remainder when 111 is divided by 5? (a) 2 (b) 0 (c) 4 (d) 1
27. 27.What is the smallest number you must add to 427 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 5
28. 28.What is the smallest number you must add to 229 to make it divisible by 4? (a) 1 (b) 2 (c) 3
29. 29.What is the units digit of 3^23? (a) 7 (b) 9 (c) 3 (d) 1
30. 30.What is the remainder when 602 is divided by 10? (a) 1 (b) 3 (c) 8 (d) 2
31. 31.Which of these numbers is divisible by 8? (a) 4712 (b) 4727 (c) 4708 (d) 4718
32. 32.What is the smallest number you must add to 59 to make it divisible by 3? (a) 1 (b) 2
33. 33.What is the units digit of 3^16? (a) 3 (b) 7 (c) 9 (d) 1
34. 34.Which of these numbers is divisible by 6? (a) 4253 (b) 4242 (c) 4244 (d) 4241
35. 35.What is the smallest number you must add to 519 to make it divisible by 10? (a) 9 (b) 1 (c) 2 (d) 3
36. 36.What is the units digit of 3^22? (a) 7 (b) 9 (c) 1 (d) 3
37. 37.Which of these numbers is divisible by 9? (a) 1405 (b) 1420 (c) 1428 (d) 1413
38. 38.Which of these numbers is divisible by 8? (a) 2163 (b) 2152 (c) 2154 (d) 2149
39. 39.Which of these numbers is divisible by 8? (a) 5377 (b) 5386 (c) 5384 (d) 5399
40. 40.What is the remainder when 643 is divided by 8? (a) 2 (b) 3 (c) 4 (d) 5

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