Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 847 · 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 units digit of 2^16? (a) 8 (b) 4 (c) 2 (d) 6
2. 2.Which of these numbers is divisible by 9? (a) 4996 (b) 4994 (c) 5005 (d) 4995
3. 3.What is the remainder when 157 is divided by 11? (a) 2 (b) 8 (c) 3 (d) 4
4. 4.What is the smallest number you must add to 208 to make it divisible by 6? (a) 3 (b) 2 (c) 4 (d) 1
5. 5.Which of these numbers is divisible by 3? (a) 1236 (b) 1240 (c) 1235 (d) 1238
6. 6.What is the remainder when 215 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 5
7. 7.What is the units digit of 2^19? (a) 6 (b) 4 (c) 8 (d) 2
8. 8.What is the units digit of 9^35? (a) 1 (b) 8 (c) 7 (d) 9
9. 9.What is the units digit of 2^49? (a) 6 (b) 2 (c) 8 (d) 4
10. 10.Which of these numbers is divisible by 4? (a) 2961 (b) 2960 (c) 2959 (d) 2965
11. 11.What is the smallest number you must add to 161 to make it divisible by 8? (a) 5 (b) 6 (c) 1 (d) 7
12. 12.Which of these numbers is divisible by 11? (a) 3044 (b) 3055 (c) 3036 (d) 3029
13. 13.What is the remainder when 109 is divided by 3? (a) 2 (b) 0 (c) 1
14. 14.Which of these numbers is divisible by 8? (a) 6053 (b) 6046 (c) 6034 (d) 6040
15. 15.What is the units digit of 8^13? (a) 4 (b) 2 (c) 8 (d) 6
16. 16.What is the smallest number you must add to 137 to make it divisible by 10? (a) 3 (b) 4 (c) 7 (d) 2
17. 17.What is the remainder when 886 is divided by 11? (a) 7 (b) 8 (c) 5 (d) 6
18. 18.What is the smallest number you must add to 438 to make it divisible by 7? (a) 5 (b) 3 (c) 2 (d) 4
19. 19.What is the units digit of 3^22? (a) 3 (b) 7 (c) 1 (d) 9
20. 20.What is the smallest number you must add to 207 to make it divisible by 8? (a) 7 (b) 2 (c) 1 (d) 3
21. 21.What is the remainder when 215 is divided by 7? (a) 5 (b) 6 (c) 2 (d) 4
22. 22.What is the smallest number you must add to 506 to make it divisible by 8? (a) 2 (b) 6 (c) 7 (d) 5
23. 23.What is the remainder when 146 is divided by 7? (a) 5 (b) 4 (c) 1 (d) 6
24. 24.What is the units digit of 3^14? (a) 1 (b) 9 (c) 7 (d) 3
25. 25.What is the units digit of 9^23? (a) 7 (b) 9 (c) 1 (d) 8
26. 26.What is the units digit of 4^44? (a) 4 (b) 6 (c) 5 (d) 7
27. 27.What is the remainder when 232 is divided by 12? (a) 8 (b) 4 (c) 5 (d) 3
28. 28.What is the smallest number you must add to 111 to make it divisible by 4? (a) 1 (b) 3 (c) 2
29. 29.Which of these numbers is divisible by 11? (a) 6354 (b) 6366 (c) 6347 (d) 6340
30. 30.What is the smallest number you must add to 79 to make it divisible by 7? (a) 2 (b) 6 (c) 5 (d) 4
31. 31.What is the remainder when 276 is divided by 7? (a) 2 (b) 5 (c) 4 (d) 3
32. 32.What is the units digit of 8^14? (a) 6 (b) 4 (c) 2 (d) 8
33. 33.Which of these numbers is divisible by 3? (a) 2330 (b) 2327 (c) 2328 (d) 2332
34. 34.Which of these numbers is divisible by 6? (a) 1456 (b) 1445 (c) 1446 (d) 1448
35. 35.What is the remainder when 230 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
36. 36.What is the units digit of 2^23? (a) 8 (b) 4 (c) 2 (d) 6
37. 37.What is the smallest number you must add to 254 to make it divisible by 10? (a) 6 (b) 4 (c) 5 (d) 7
38. 38.Which of these numbers is divisible by 4? (a) 3298 (b) 3306 (c) 3303 (d) 3300
39. 39.Which of these numbers is divisible by 11? (a) 5637 (b) 5643 (c) 5653 (d) 5657
40. 40.What is the remainder when 329 is divided by 9? (a) 7 (b) 5 (c) 6 (d) 4

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