Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 104 · 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) 3440 (b) 3450 (c) 3439 (d) 3444
2. 2.Which of these numbers is divisible by 4? (a) 747 (b) 740 (c) 742 (d) 739
3. 3.What is the remainder when 65 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
4. 4.Which of these numbers is divisible by 3? (a) 922 (b) 925 (c) 924 (d) 928
5. 5.Which of these numbers is divisible by 3? (a) 2338 (b) 2332 (c) 2334 (d) 2336
6. 6.What is the remainder when 664 is divided by 9? (a) 2 (b) 6 (c) 8 (d) 7
7. 7.What is the remainder when 109 is divided by 10? (a) 7 (b) 8 (c) 1 (d) 9
8. 8.What is the remainder when 363 is divided by 9? (a) 2 (b) 6 (c) 3 (d) 4
9. 9.What is the smallest number you must add to 446 to make it divisible by 11? (a) 6 (b) 5 (c) 4 (d) 7
10. 10.Which of these numbers is divisible by 9? (a) 6455 (b) 6453 (c) 6463 (d) 6446
11. 11.What is the smallest number you must add to 216 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
12. 12.What is the units digit of 8^59? (a) 4 (b) 2 (c) 6 (d) 8
13. 13.Which of these numbers is divisible by 6? (a) 5088 (b) 5097 (c) 5089 (d) 5085
14. 14.What is the smallest number you must add to 143 to make it divisible by 4? (a) 3 (b) 2 (c) 1
15. 15.Which of these numbers is divisible by 11? (a) 1448 (b) 1421 (c) 1435 (d) 1430
16. 16.What is the smallest number you must add to 897 to make it divisible by 12? (a) 4 (b) 2 (c) 3 (d) 9
17. 17.Which of these numbers is divisible by 11? (a) 8455 (b) 8440 (c) 8448 (d) 8460
18. 18.What is the remainder when 79 is divided by 7? (a) 1 (b) 2 (c) 3 (d) 5
19. 19.What is the smallest number you must add to 88 to make it divisible by 6? (a) 2 (b) 3 (c) 1 (d) 4
20. 20.Which of these numbers is divisible by 8? (a) 3384 (b) 3381 (c) 3385 (d) 3394
21. 21.What is the units digit of 3^51? (a) 7 (b) 9 (c) 1 (d) 3
22. 22.Which of these numbers is divisible by 8? (a) 1717 (b) 1710 (c) 1712 (d) 1724
23. 23.What is the remainder when 40 is divided by 3? (a) 1 (b) 0 (c) 2
24. 24.What is the smallest number you must add to 922 to make it divisible by 12? (a) 3 (b) 1 (c) 10 (d) 2
25. 25.Which of these numbers is divisible by 6? (a) 1927 (b) 1932 (c) 1943 (d) 1937
26. 26.What is the units digit of 4^12? (a) 5 (b) 4 (c) 7 (d) 6
27. 27.Which of these numbers is divisible by 4? (a) 1000 (b) 1003 (c) 997 (d) 1005
28. 28.What is the smallest number you must add to 398 to make it divisible by 12? (a) 11 (b) 9 (c) 10 (d) 2
29. 29.What is the remainder when 426 is divided by 9? (a) 6 (b) 3 (c) 2 (d) 4
30. 30.What is the remainder when 525 is divided by 8? (a) 3 (b) 6 (c) 5 (d) 4
31. 31.Which of these numbers is divisible by 4? (a) 462 (b) 453 (c) 457 (d) 456
32. 32.What is the units digit of 9^23? (a) 9 (b) 1 (c) 7 (d) 8
33. 33.What is the units digit of 7^29? (a) 3 (b) 7 (c) 1 (d) 9
34. 34.Which of these numbers is divisible by 3? (a) 2371 (b) 2375 (c) 2373 (d) 2377
35. 35.What is the remainder when 420 is divided by 9? (a) 3 (b) 6 (c) 7 (d) 5
36. 36.What is the units digit of 7^50? (a) 7 (b) 1 (c) 3 (d) 9
37. 37.What is the units digit of 3^16? (a) 7 (b) 9 (c) 3 (d) 1
38. 38.Which of these numbers is divisible by 3? (a) 350 (b) 355 (c) 351 (d) 352
39. 39.What is the remainder when 139 is divided by 3? (a) 0 (b) 2 (c) 1
40. 40.What is the remainder when 62 is divided by 3? (a) 1 (b) 2 (c) 0

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