Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 134 · 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 216 to make it divisible by 10? (a) 3 (b) 5 (c) 4 (d) 6
2. 2.What is the remainder when 361 is divided by 8? (a) 0 (b) 2 (c) 1 (d) 7
3. 3.What is the remainder when 304 is divided by 9? (a) 7 (b) 8 (c) 6 (d) 2
4. 4.Which of these numbers is divisible by 6? (a) 4179 (b) 4170 (c) 4175 (d) 4169
5. 5.What is the units digit of 2^15? (a) 6 (b) 2 (c) 8 (d) 4
6. 6.What is the smallest number you must add to 228 to make it divisible by 9? (a) 6 (b) 5 (c) 3 (d) 7
7. 7.Which of these numbers is divisible by 3? (a) 517 (b) 520 (c) 516 (d) 514
8. 8.What is the remainder when 47 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
9. 9.What is the units digit of 2^49? (a) 8 (b) 6 (c) 2 (d) 4
10. 10.What is the units digit of 4^50? (a) 5 (b) 7 (c) 6 (d) 4
11. 11.What is the remainder when 474 is divided by 8? (a) 2 (b) 3 (c) 6 (d) 1
12. 12.What is the units digit of 7^40? (a) 1 (b) 7 (c) 9 (d) 3
13. 13.Which of these numbers is divisible by 8? (a) 523 (b) 520 (c) 515 (d) 533
14. 14.What is the remainder when 394 is divided by 7? (a) 1 (b) 2 (c) 3 (d) 5
15. 15.What is the remainder when 115 is divided by 3? (a) 2 (b) 0 (c) 1
16. 16.Which of these numbers is divisible by 3? (a) 1580 (b) 1583 (c) 1586 (d) 1581
17. 17.What is the remainder when 622 is divided by 10? (a) 8 (b) 2 (c) 3 (d) 1
18. 18.What is the units digit of 2^36? (a) 2 (b) 8 (c) 6 (d) 4
19. 19.What is the units digit of 9^60? (a) 9 (b) 1 (c) 0 (d) 2
20. 20.What is the remainder when 371 is divided by 5? (a) 0 (b) 4 (c) 2 (d) 1
21. 21.What is the units digit of 4^25? (a) 6 (b) 5 (c) 4 (d) 3
22. 22.What is the smallest number you must add to 316 to make it divisible by 6? (a) 1 (b) 3 (c) 2 (d) 4
23. 23.Which of these numbers is divisible by 11? (a) 9137 (b) 9119 (c) 9124 (d) 9117
24. 24.What is the remainder when 386 is divided by 8? (a) 6 (b) 3 (c) 1 (d) 2
25. 25.Which of these numbers is divisible by 3? (a) 665 (b) 671 (c) 666 (d) 667
26. 26.What is the smallest number you must add to 603 to make it divisible by 10? (a) 3 (b) 6 (c) 7 (d) 8
27. 27.What is the remainder when 199 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
28. 28.What is the remainder when 159 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
29. 29.What is the smallest number you must add to 133 to make it divisible by 4? (a) 2 (b) 1 (c) 3
30. 30.Which of these numbers is divisible by 9? (a) 6332 (b) 6348 (c) 6337 (d) 6336
31. 31.What is the smallest number you must add to 97 to make it divisible by 4? (a) 1 (b) 3 (c) 2
32. 32.Which of these numbers is divisible by 6? (a) 4020 (b) 4023 (c) 4028 (d) 4017
33. 33.Which of these numbers is divisible by 6? (a) 3117 (b) 3111 (c) 3108 (d) 3106
34. 34.What is the remainder when 187 is divided by 7? (a) 6 (b) 2 (c) 4 (d) 5
35. 35.Which of these numbers is divisible by 6? (a) 3544 (b) 3540 (c) 3535 (d) 3548
36. 36.What is the units digit of 2^30? (a) 2 (b) 4 (c) 8 (d) 6
37. 37.Which of these numbers is divisible by 11? (a) 7036 (b) 7028 (c) 7009 (d) 7018
38. 38.What is the remainder when 501 is divided by 8? (a) 5 (b) 6 (c) 3 (d) 4
39. 39.What is the smallest number you must add to 139 to make it divisible by 4? (a) 3 (b) 1 (c) 2
40. 40.Which of these numbers is divisible by 3? (a) 486 (b) 487 (c) 484 (d) 490

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