Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 262 · 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 remainder when 425 is divided by 12? (a) 7 (b) 6 (c) 4 (d) 5
2. 2.Which of these numbers is divisible by 4? (a) 1266 (b) 1269 (c) 1262 (d) 1264
3. 3.Which of these numbers is divisible by 6? (a) 1512 (b) 1511 (c) 1516 (d) 1523
4. 4.What is the remainder when 469 is divided by 12? (a) 0 (b) 2 (c) 1 (d) 11
5. 5.What is the units digit of 4^40? (a) 7 (b) 4 (c) 5 (d) 6
6. 6.What is the smallest number you must add to 184 to make it divisible by 9? (a) 7 (b) 6 (c) 4 (d) 5
7. 7.Which of these numbers is divisible by 3? (a) 2299 (b) 2296 (c) 2293 (d) 2295
8. 8.What is the smallest number you must add to 440 to make it divisible by 7? (a) 1 (b) 3 (c) 6 (d) 2
9. 9.What is the smallest number you must add to 218 to make it divisible by 7? (a) 6 (b) 4 (c) 1 (d) 5
10. 10.What is the units digit of 8^25? (a) 6 (b) 8 (c) 2 (d) 4
11. 11.What is the remainder when 175 is divided by 3? (a) 2 (b) 0 (c) 1
12. 12.Which of these numbers is divisible by 11? (a) 1855 (b) 1875 (c) 1864 (d) 1859
13. 13.What is the smallest number you must add to 190 to make it divisible by 7? (a) 4 (b) 6 (c) 5 (d) 1
14. 14.What is the units digit of 2^12? (a) 2 (b) 8 (c) 4 (d) 6
15. 15.Which of these numbers is divisible by 3? (a) 625 (b) 632 (c) 628 (d) 627
16. 16.Which of these numbers is divisible by 3? (a) 2057 (b) 2050 (c) 2054 (d) 2052
17. 17.What is the smallest number you must add to 548 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 2
18. 18.What is the remainder when 211 is divided by 11? (a) 1 (b) 2 (c) 3 (d) 9
19. 19.What is the remainder when 109 is divided by 3? (a) 1 (b) 0 (c) 2
20. 20.What is the remainder when 290 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
21. 21.What is the remainder when 267 is divided by 6? (a) 3 (b) 2 (c) 5 (d) 4
22. 22.What is the smallest number you must add to 267 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
23. 23.Which of these numbers is divisible by 9? (a) 5016 (b) 5023 (c) 5013 (d) 5007
24. 24.What is the units digit of 7^52? (a) 7 (b) 1 (c) 9 (d) 3
25. 25.Which of these numbers is divisible by 3? (a) 612 (b) 613 (c) 610 (d) 617
26. 26.What is the units digit of 9^20? (a) 2 (b) 1 (c) 0 (d) 9
27. 27.What is the units digit of 7^60? (a) 7 (b) 3 (c) 1 (d) 9
28. 28.What is the smallest number you must add to 200 to make it divisible by 3? (a) 1 (b) 2
29. 29.What is the smallest number you must add to 422 to make it divisible by 6? (a) 4 (b) 5 (c) 2 (d) 3
30. 30.Which of these numbers is divisible by 3? (a) 517 (b) 521 (c) 524 (d) 519
31. 31.What is the remainder when 94 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
32. 32.What is the units digit of 2^35? (a) 2 (b) 8 (c) 4 (d) 6
33. 33.Which of these numbers is divisible by 8? (a) 6338 (b) 6328 (c) 6331 (d) 6321
34. 34.What is the remainder when 626 is divided by 12? (a) 3 (b) 2 (c) 10 (d) 1
35. 35.What is the smallest number you must add to 275 to make it divisible by 4? (a) 3 (b) 2 (c) 1
36. 36.What is the units digit of 8^19? (a) 4 (b) 8 (c) 6 (d) 2
37. 37.Which of these numbers is divisible by 3? (a) 2691 (b) 2690 (c) 2696 (d) 2693
38. 38.What is the remainder when 208 is divided by 11? (a) 1 (b) 9 (c) 8 (d) 10
39. 39.Which of these numbers is divisible by 8? (a) 615 (b) 630 (c) 621 (d) 616
40. 40.Which of these numbers is divisible by 11? (a) 4791 (b) 4804 (c) 4815 (d) 4796

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