Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 804 · 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 6? (a) 4537 (b) 4534 (c) 4528 (d) 4530
2. 2.What is the units digit of 3^15? (a) 3 (b) 7 (c) 1 (d) 9
3. 3.What is the smallest number you must add to 205 to make it divisible by 3? (a) 1 (b) 2
4. 4.Which of these numbers is divisible by 3? (a) 2577 (b) 2581 (c) 2576 (d) 2579
5. 5.Which of these numbers is divisible by 8? (a) 5974 (b) 5953 (c) 5960 (d) 5967
6. 6.What is the remainder when 97 is divided by 3? (a) 2 (b) 1 (c) 0
7. 7.What is the units digit of 2^14? (a) 8 (b) 2 (c) 6 (d) 4
8. 8.What is the units digit of 7^34? (a) 1 (b) 7 (c) 9 (d) 3
9. 9.What is the remainder when 314 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
10. 10.What is the remainder when 69 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
11. 11.Which of these numbers is divisible by 11? (a) 7864 (b) 7854 (c) 7866 (d) 7849
12. 12.What is the smallest number you must add to 96 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
13. 13.Which of these numbers is divisible by 4? (a) 2705 (b) 2709 (c) 2701 (d) 2704
14. 14.What is the smallest number you must add to 242 to make it divisible by 7? (a) 5 (b) 3 (c) 2 (d) 4
15. 15.Which of these numbers is divisible by 4? (a) 3506 (b) 3514 (c) 3510 (d) 3508
16. 16.Which of these numbers is divisible by 3? (a) 153 (b) 157 (c) 152 (d) 155
17. 17.Which of these numbers is divisible by 4? (a) 761 (b) 754 (c) 756 (d) 759
18. 18.What is the units digit of 3^22? (a) 7 (b) 3 (c) 1 (d) 9
19. 19.What is the smallest number you must add to 397 to make it divisible by 7? (a) 5 (b) 1 (c) 2 (d) 3
20. 20.Which of these numbers is divisible by 9? (a) 6102 (b) 6112 (c) 6097 (d) 6106
21. 21.What is the remainder when 438 is divided by 11? (a) 10 (b) 9 (c) 2 (d) 8
22. 22.Which of these numbers is divisible by 3? (a) 2452 (b) 2448 (c) 2449 (d) 2447
23. 23.Which of these numbers is divisible by 8? (a) 6428 (b) 6423 (c) 6413 (d) 6416
24. 24.What is the remainder when 104 is divided by 3? (a) 2 (b) 1 (c) 0
25. 25.Which of these numbers is divisible by 8? (a) 5150 (b) 5144 (c) 5143 (d) 5153
26. 26.What is the units digit of 4^55? (a) 4 (b) 5 (c) 3 (d) 6
27. 27.What is the units digit of 9^19? (a) 1 (b) 7 (c) 8 (d) 9
28. 28.What is the smallest number you must add to 565 to make it divisible by 7? (a) 1 (b) 3 (c) 2 (d) 5
29. 29.What is the remainder when 62 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
30. 30.What is the units digit of 9^46? (a) 2 (b) 0 (c) 9 (d) 1
31. 31.What is the smallest number you must add to 131 to make it divisible by 3? (a) 1 (b) 2
32. 32.Which of these numbers is divisible by 11? (a) 585 (b) 577 (c) 572 (d) 564
33. 33.What is the units digit of 2^13? (a) 4 (b) 2 (c) 8 (d) 6
34. 34.What is the remainder when 288 is divided by 10? (a) 9 (b) 8 (c) 2 (d) 7
35. 35.What is the units digit of 2^18? (a) 8 (b) 6 (c) 4 (d) 2
36. 36.What is the remainder when 219 is divided by 8? (a) 3 (b) 5 (c) 4 (d) 2
37. 37.What is the remainder when 514 is divided by 12? (a) 9 (b) 11 (c) 10 (d) 2
38. 38.What is the units digit of 3^41? (a) 9 (b) 7 (c) 3 (d) 1
39. 39.Which of these numbers is divisible by 4? (a) 1155 (b) 1145 (c) 1148 (d) 1151
40. 40.What is the units digit of 7^14? (a) 3 (b) 1 (c) 9 (d) 7

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