Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 682 · 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 506 to make it divisible by 7? (a) 4 (b) 6 (c) 2 (d) 5
2. 2.What is the remainder when 304 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
3. 3.What is the units digit of 7^59? (a) 1 (b) 7 (c) 9 (d) 3
4. 4.What is the units digit of 3^58? (a) 1 (b) 7 (c) 3 (d) 9
5. 5.Which of these numbers is divisible by 9? (a) 6327 (b) 6324 (c) 6343 (d) 6334
6. 6.Which of these numbers is divisible by 11? (a) 7003 (b) 6996 (c) 6987 (d) 7017
7. 7.What is the units digit of 3^35? (a) 9 (b) 1 (c) 3 (d) 7
8. 8.What is the smallest number you must add to 508 to make it divisible by 9? (a) 4 (b) 5 (c) 6 (d) 7
9. 9.What is the smallest number you must add to 287 to make it divisible by 4? (a) 3 (b) 1 (c) 2
10. 10.What is the units digit of 7^12? (a) 1 (b) 9 (c) 3 (d) 7
11. 11.What is the units digit of 4^53? (a) 6 (b) 3 (c) 5 (d) 4
12. 12.What is the units digit of 2^22? (a) 4 (b) 6 (c) 8 (d) 2
13. 13.Which of these numbers is divisible by 11? (a) 6616 (b) 6622 (c) 6642 (d) 6626
14. 14.What is the remainder when 127 is divided by 3? (a) 2 (b) 0 (c) 1
15. 15.What is the units digit of 9^22? (a) 1 (b) 9 (c) 2 (d) 0
16. 16.Which of these numbers is divisible by 11? (a) 2122 (b) 2123 (c) 2137 (d) 2128
17. 17.What is the smallest number you must add to 419 to make it divisible by 9? (a) 4 (b) 3 (c) 6 (d) 5
18. 18.Which of these numbers is divisible by 3? (a) 2584 (b) 2591 (c) 2588 (d) 2586
19. 19.What is the smallest number you must add to 745 to make it divisible by 11? (a) 8 (b) 4 (c) 3 (d) 2
20. 20.What is the remainder when 656 is divided by 9? (a) 6 (b) 7 (c) 8 (d) 1
21. 21.Which of these numbers is divisible by 11? (a) 1427 (b) 1430 (c) 1443 (d) 1432
22. 22.What is the smallest number you must add to 122 to make it divisible by 3? (a) 1 (b) 2
23. 23.Which of these numbers is divisible by 4? (a) 1121 (b) 1120 (c) 1117 (d) 1125
24. 24.Which of these numbers is divisible by 6? (a) 4852 (b) 4844 (c) 4848 (d) 4856
25. 25.What is the smallest number you must add to 928 to make it divisible by 12? (a) 9 (b) 8 (c) 7 (d) 4
26. 26.What is the remainder when 361 is divided by 10? (a) 9 (b) 2 (c) 0 (d) 1
27. 27.Which of these numbers is divisible by 9? (a) 7280 (b) 7269 (c) 7272 (d) 7286
28. 28.Which of these numbers is divisible by 4? (a) 1813 (b) 1817 (c) 1816 (d) 1823
29. 29.What is the remainder when 224 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
30. 30.What is the smallest number you must add to 138 to make it divisible by 4? (a) 2 (b) 1 (c) 3
31. 31.What is the smallest number you must add to 222 to make it divisible by 11? (a) 10 (b) 2 (c) 9 (d) 8
32. 32.What is the units digit of 2^29? (a) 8 (b) 2 (c) 4 (d) 6
33. 33.What is the smallest number you must add to 592 to make it divisible by 12? (a) 7 (b) 8 (c) 9 (d) 4
34. 34.What is the smallest number you must add to 554 to make it divisible by 10? (a) 4 (b) 5 (c) 7 (d) 6
35. 35.What is the units digit of 9^52? (a) 9 (b) 0 (c) 1 (d) 2
36. 36.Which of these numbers is divisible by 11? (a) 6218 (b) 6215 (c) 6233 (d) 6205
37. 37.Which of these numbers is divisible by 3? (a) 1406 (b) 1401 (c) 1402 (d) 1400
38. 38.What is the smallest number you must add to 99 to make it divisible by 4? (a) 2 (b) 1 (c) 3
39. 39.What is the remainder when 301 is divided by 6? (a) 0 (b) 5 (c) 1 (d) 2
40. 40.Which of these numbers is divisible by 3? (a) 2089 (b) 2085 (c) 2087 (d) 2083

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