Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 450 · 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 400 to make it divisible by 6? (a) 3 (b) 1 (c) 2 (d) 4
2. 2.Which of these numbers is divisible by 4? (a) 375 (b) 378 (c) 382 (d) 376
3. 3.What is the units digit of 8^51? (a) 4 (b) 2 (c) 8 (d) 6
4. 4.What is the remainder when 140 is divided by 12? (a) 7 (b) 4 (c) 9 (d) 8
5. 5.Which of these numbers is divisible by 6? (a) 722 (b) 727 (c) 726 (d) 735
6. 6.What is the units digit of 3^25? (a) 7 (b) 1 (c) 3 (d) 9
7. 7.What is the units digit of 2^22? (a) 6 (b) 4 (c) 8 (d) 2
8. 8.What is the units digit of 8^52? (a) 6 (b) 2 (c) 8 (d) 4
9. 9.What is the smallest number you must add to 226 to make it divisible by 3? (a) 1 (b) 2
10. 10.Which of these numbers is divisible by 9? (a) 7416 (b) 7426 (c) 7408 (d) 7421
11. 11.What is the remainder when 611 is divided by 8? (a) 3 (b) 5 (c) 4 (d) 2
12. 12.Which of these numbers is divisible by 4? (a) 3358 (b) 3352 (c) 3349 (d) 3355
13. 13.Which of these numbers is divisible by 8? (a) 1408 (b) 1406 (c) 1418 (d) 1415
14. 14.What is the remainder when 287 is divided by 9? (a) 6 (b) 1 (c) 8 (d) 7
15. 15.What is the smallest number you must add to 352 to make it divisible by 7? (a) 4 (b) 2 (c) 5 (d) 6
16. 16.Which of these numbers is divisible by 8? (a) 5182 (b) 5176 (c) 5175 (d) 5187
17. 17.What is the units digit of 3^22? (a) 9 (b) 7 (c) 3 (d) 1
18. 18.What is the smallest number you must add to 379 to make it divisible by 12? (a) 6 (b) 5 (c) 4 (d) 7
19. 19.What is the units digit of 4^12? (a) 4 (b) 5 (c) 7 (d) 6
20. 20.What is the smallest number you must add to 107 to make it divisible by 9? (a) 1 (b) 2 (c) 3 (d) 8
21. 21.What is the remainder when 131 is divided by 3? (a) 2 (b) 1 (c) 0
22. 22.What is the units digit of 2^36? (a) 2 (b) 8 (c) 4 (d) 6
23. 23.Which of these numbers is divisible by 8? (a) 528 (b) 538 (c) 534 (d) 521
24. 24.What is the units digit of 3^47? (a) 7 (b) 9 (c) 3 (d) 1
25. 25.What is the remainder when 351 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 0
26. 26.What is the units digit of 4^37? (a) 6 (b) 3 (c) 5 (d) 4
27. 27.What is the remainder when 151 is divided by 9? (a) 8 (b) 2 (c) 6 (d) 7
28. 28.What is the smallest number you must add to 178 to make it divisible by 6? (a) 1 (b) 2 (c) 3 (d) 4
29. 29.What is the remainder when 131 is divided by 6? (a) 1 (b) 3 (c) 4 (d) 5
30. 30.What is the smallest number you must add to 247 to make it divisible by 4? (a) 2 (b) 3 (c) 1
31. 31.What is the remainder when 158 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
32. 32.Which of these numbers is divisible by 9? (a) 3324 (b) 3319 (c) 3310 (d) 3312
33. 33.What is the smallest number you must add to 31 to make it divisible by 3? (a) 1 (b) 2
34. 34.Which of these numbers is divisible by 6? (a) 1129 (b) 1135 (c) 1128 (d) 1123
35. 35.Which of these numbers is divisible by 11? (a) 9837 (b) 9851 (c) 9845 (d) 9863
36. 36.What is the remainder when 410 is divided by 9? (a) 4 (b) 5 (c) 6 (d) 7
37. 37.What is the units digit of 3^24? (a) 1 (b) 3 (c) 7 (d) 9
38. 38.What is the units digit of 7^21? (a) 9 (b) 3 (c) 7 (d) 1
39. 39.What is the smallest number you must add to 45 to make it divisible by 4? (a) 2 (b) 1 (c) 3
40. 40.What is the remainder when 425 is divided by 9? (a) 7 (b) 3 (c) 1 (d) 2

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