Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 509 · 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 163 to make it divisible by 4? (a) 2 (b) 1 (c) 3
2. 2.Which of these numbers is divisible by 3? (a) 362 (b) 355 (c) 357 (d) 358
3. 3.Which of these numbers is divisible by 11? (a) 7035 (b) 7029 (c) 7045 (d) 7019
4. 4.Which of these numbers is divisible by 4? (a) 1814 (b) 1808 (c) 1809 (d) 1806
5. 5.Which of these numbers is divisible by 11? (a) 4386 (b) 4389 (c) 4409 (d) 4394
6. 6.What is the smallest number you must add to 538 to make it divisible by 8? (a) 6 (b) 5 (c) 7 (d) 2
7. 7.What is the units digit of 8^29? (a) 4 (b) 8 (c) 2 (d) 6
8. 8.What is the remainder when 146 is divided by 11? (a) 4 (b) 3 (c) 8 (d) 2
9. 9.Which of these numbers is divisible by 9? (a) 6198 (b) 6208 (c) 6201 (d) 6214
10. 10.Which of these numbers is divisible by 3? (a) 2418 (b) 2416 (c) 2423 (d) 2419
11. 11.What is the smallest number you must add to 347 to make it divisible by 12? (a) 11 (b) 2 (c) 1 (d) 3
12. 12.Which of these numbers is divisible by 4? (a) 2656 (b) 2655 (c) 2662 (d) 2657
13. 13.Which of these numbers is divisible by 6? (a) 2083 (b) 2080 (c) 2091 (d) 2082
14. 14.What is the units digit of 7^56? (a) 1 (b) 9 (c) 3 (d) 7
15. 15.What is the units digit of 8^30? (a) 4 (b) 8 (c) 2 (d) 6
16. 16.What is the smallest number you must add to 163 to make it divisible by 3? (a) 2 (b) 1
17. 17.What is the units digit of 3^31? (a) 9 (b) 3 (c) 7 (d) 1
18. 18.Which of these numbers is divisible by 11? (a) 3052 (b) 3047 (c) 3059 (d) 3040
19. 19.What is the remainder when 221 is divided by 9? (a) 7 (b) 4 (c) 6 (d) 5
20. 20.What is the units digit of 4^23? (a) 5 (b) 4 (c) 6 (d) 3
21. 21.What is the smallest number you must add to 711 to make it divisible by 11? (a) 5 (b) 3 (c) 7 (d) 4
22. 22.What is the smallest number you must add to 68 to make it divisible by 6? (a) 3 (b) 2 (c) 5 (d) 4
23. 23.What is the units digit of 4^27? (a) 5 (b) 4 (c) 3 (d) 6
24. 24.Which of these numbers is divisible by 3? (a) 496 (b) 494 (c) 492 (d) 491
25. 25.Which of these numbers is divisible by 8? (a) 3824 (b) 3836 (c) 3828 (d) 3819
26. 26.What is the smallest number you must add to 656 to make it divisible by 9? (a) 3 (b) 8 (c) 2 (d) 1
27. 27.What is the remainder when 527 is divided by 10? (a) 8 (b) 3 (c) 7 (d) 6
28. 28.Which of these numbers is divisible by 6? (a) 5140 (b) 5136 (c) 5147 (d) 5135
29. 29.What is the units digit of 7^40? (a) 1 (b) 3 (c) 7 (d) 9
30. 30.What is the units digit of 8^14? (a) 8 (b) 6 (c) 2 (d) 4
31. 31.What is the units digit of 8^35? (a) 6 (b) 4 (c) 2 (d) 8
32. 32.Which of these numbers is divisible by 8? (a) 3582 (b) 3584 (c) 3599 (d) 3586
33. 33.What is the remainder when 118 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
34. 34.Which of these numbers is divisible by 9? (a) 3429 (b) 3439 (c) 3423 (d) 3432
35. 35.What is the remainder when 231 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
36. 36.What is the remainder when 275 is divided by 7? (a) 1 (b) 2 (c) 5 (d) 3
37. 37.What is the remainder when 153 is divided by 7? (a) 1 (b) 5 (c) 6 (d) 4
38. 38.What is the smallest number you must add to 247 to make it divisible by 12? (a) 7 (b) 6 (c) 4 (d) 5
39. 39.Which of these numbers is divisible by 4? (a) 2718 (b) 2714 (c) 2711 (d) 2712
40. 40.What is the remainder when 588 is divided by 9? (a) 4 (b) 6 (c) 3 (d) 2

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