Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 364 · 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 8? (a) 3443 (b) 3430 (c) 3432 (d) 3437
2. 2.What is the units digit of 2^17? (a) 2 (b) 6 (c) 4 (d) 8
3. 3.What is the smallest number you must add to 403 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
4. 4.What is the remainder when 119 is divided by 3? (a) 1 (b) 2 (c) 0
5. 5.What is the units digit of 7^35? (a) 9 (b) 7 (c) 1 (d) 3
6. 6.Which of these numbers is divisible by 4? (a) 322 (b) 313 (c) 316 (d) 317
7. 7.What is the remainder when 157 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
8. 8.What is the remainder when 213 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
9. 9.What is the smallest number you must add to 239 to make it divisible by 10? (a) 3 (b) 2 (c) 9 (d) 1
10. 10.What is the units digit of 3^53? (a) 7 (b) 9 (c) 3 (d) 1
11. 11.What is the smallest number you must add to 138 to make it divisible by 7? (a) 3 (b) 2 (c) 1 (d) 5
12. 12.What is the remainder when 407 is divided by 8? (a) 1 (b) 7 (c) 5 (d) 6
13. 13.Which of these numbers is divisible by 6? (a) 1643 (b) 1646 (c) 1638 (d) 1633
14. 14.Which of these numbers is divisible by 11? (a) 9289 (b) 9284 (c) 9299 (d) 9278
15. 15.What is the smallest number you must add to 182 to make it divisible by 6? (a) 2 (b) 4 (c) 5 (d) 3
16. 16.What is the units digit of 7^34? (a) 3 (b) 1 (c) 7 (d) 9
17. 17.Which of these numbers is divisible by 6? (a) 4147 (b) 4142 (c) 4146 (d) 4154
18. 18.What is the remainder when 457 is divided by 12? (a) 0 (b) 2 (c) 11 (d) 1
19. 19.What is the units digit of 4^44? (a) 4 (b) 5 (c) 7 (d) 6
20. 20.Which of these numbers is divisible by 3? (a) 1508 (b) 1502 (c) 1503 (d) 1504
21. 21.What is the units digit of 7^48? (a) 3 (b) 7 (c) 1 (d) 9
22. 22.What is the smallest number you must add to 517 to make it divisible by 7? (a) 3 (b) 1 (c) 6 (d) 2
23. 23.What is the units digit of 7^27? (a) 7 (b) 9 (c) 3 (d) 1
24. 24.What is the remainder when 303 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
25. 25.What is the remainder when 218 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
26. 26.Which of these numbers is divisible by 3? (a) 1663 (b) 1661 (c) 1658 (d) 1659
27. 27.Which of these numbers is divisible by 11? (a) 614 (b) 616 (c) 623 (d) 635
28. 28.Which of these numbers is divisible by 9? (a) 3034 (b) 3045 (c) 3042 (d) 3054
29. 29.What is the smallest number you must add to 354 to make it divisible by 8? (a) 2 (b) 5 (c) 6 (d) 7
30. 30.Which of these numbers is divisible by 3? (a) 261 (b) 266 (c) 259 (d) 262
31. 31.What is the units digit of 8^18? (a) 2 (b) 6 (c) 8 (d) 4
32. 32.What is the remainder when 466 is divided by 9? (a) 8 (b) 2 (c) 6 (d) 7
33. 33.What is the smallest number you must add to 214 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
34. 34.Which of these numbers is divisible by 4? (a) 1569 (b) 1575 (c) 1568 (d) 1567
35. 35.What is the smallest number you must add to 690 to make it divisible by 9? (a) 3 (b) 4 (c) 6 (d) 2
36. 36.What is the smallest number you must add to 293 to make it divisible by 8? (a) 4 (b) 3 (c) 5 (d) 2
37. 37.What is the smallest number you must add to 442 to make it divisible by 11? (a) 9 (b) 8 (c) 2 (d) 10
38. 38.What is the smallest number you must add to 523 to make it divisible by 12? (a) 6 (b) 5 (c) 4 (d) 7
39. 39.Which of these numbers is divisible by 6? (a) 2095 (b) 2093 (c) 2104 (d) 2094
40. 40.What is the remainder when 79 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2

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