Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 733 · 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 185 is divided by 12? (a) 6 (b) 7 (c) 5 (d) 4
2. 2.What is the smallest number you must add to 341 to make it divisible by 10? (a) 1 (b) 8 (c) 7 (d) 9
3. 3.What is the smallest number you must add to 127 to make it divisible by 6? (a) 4 (b) 1 (c) 3 (d) 5
4. 4.Which of these numbers is divisible by 4? (a) 2494 (b) 2497 (c) 2496 (d) 2502
5. 5.Which of these numbers is divisible by 9? (a) 5917 (b) 5897 (c) 5908 (d) 5904
6. 6.Which of these numbers is divisible by 11? (a) 7759 (b) 7736 (c) 7750 (d) 7744
7. 7.What is the remainder when 789 is divided by 12? (a) 3 (b) 9 (c) 8 (d) 10
8. 8.What is the units digit of 9^35? (a) 7 (b) 9 (c) 1 (d) 8
9. 9.Which of these numbers is divisible by 6? (a) 3750 (b) 3760 (c) 3752 (d) 3749
10. 10.What is the smallest number you must add to 362 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
11. 11.What is the units digit of 4^10? (a) 7 (b) 6 (c) 5 (d) 4
12. 12.What is the remainder when 277 is divided by 8? (a) 3 (b) 6 (c) 5 (d) 4
13. 13.Which of these numbers is divisible by 11? (a) 8803 (b) 8816 (c) 8800 (d) 8792
14. 14.Which of these numbers is divisible by 4? (a) 1140 (b) 1141 (c) 1147 (d) 1139
15. 15.Which of these numbers is divisible by 3? (a) 657 (b) 662 (c) 656 (d) 658
16. 16.What is the remainder when 278 is divided by 9? (a) 1 (b) 8 (c) 6 (d) 7
17. 17.What is the units digit of 2^38? (a) 8 (b) 2 (c) 6 (d) 4
18. 18.What is the units digit of 3^57? (a) 3 (b) 7 (c) 9 (d) 1
19. 19.What is the smallest number you must add to 541 to make it divisible by 9? (a) 7 (b) 6 (c) 1 (d) 8
20. 20.What is the units digit of 3^33? (a) 9 (b) 1 (c) 7 (d) 3
21. 21.What is the smallest number you must add to 167 to make it divisible by 6? (a) 2 (b) 3 (c) 1 (d) 5
22. 22.Which of these numbers is divisible by 4? (a) 624 (b) 630 (c) 622 (d) 625
23. 23.What is the units digit of 7^24? (a) 7 (b) 3 (c) 9 (d) 1
24. 24.What is the smallest number you must add to 613 to make it divisible by 10? (a) 6 (b) 3 (c) 7 (d) 8
25. 25.Which of these numbers is divisible by 11? (a) 4381 (b) 4367 (c) 4361 (d) 4373
26. 26.What is the remainder when 531 is divided by 11? (a) 4 (b) 8 (c) 3 (d) 2
27. 27.What is the units digit of 8^37? (a) 4 (b) 2 (c) 8 (d) 6
28. 28.What is the units digit of 7^40? (a) 9 (b) 3 (c) 1 (d) 7
29. 29.What is the remainder when 645 is divided by 8? (a) 4 (b) 5 (c) 3 (d) 6
30. 30.What is the smallest number you must add to 205 to make it divisible by 8? (a) 4 (b) 2 (c) 3 (d) 5
31. 31.What is the remainder when 133 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
32. 32.What is the smallest number you must add to 955 to make it divisible by 12? (a) 5 (b) 6 (c) 4 (d) 7
33. 33.What is the remainder when 254 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
34. 34.Which of these numbers is divisible by 6? (a) 2842 (b) 2834 (c) 2849 (d) 2838
35. 35.Which of these numbers is divisible by 3? (a) 667 (b) 666 (c) 664 (d) 671
36. 36.What is the smallest number you must add to 233 to make it divisible by 3? (a) 2 (b) 1
37. 37.What is the remainder when 86 is divided by 5? (a) 1 (b) 0 (c) 2 (d) 4
38. 38.What is the remainder when 224 is divided by 12? (a) 9 (b) 7 (c) 4 (d) 8
39. 39.What is the remainder when 571 is divided by 10? (a) 0 (b) 9 (c) 2 (d) 1
40. 40.What is the units digit of 8^45? (a) 6 (b) 4 (c) 8 (d) 2

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