Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 717 · 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 377 is divided by 9? (a) 8 (b) 6 (c) 7 (d) 1
2. 2.What is the remainder when 619 is divided by 8? (a) 2 (b) 5 (c) 3 (d) 4
3. 3.Which of these numbers is divisible by 6? (a) 4053 (b) 4047 (c) 4058 (d) 4050
4. 4.What is the remainder when 434 is divided by 11? (a) 7 (b) 5 (c) 4 (d) 6
5. 5.What is the remainder when 257 is divided by 9? (a) 6 (b) 4 (c) 5 (d) 7
6. 6.What is the units digit of 9^50? (a) 2 (b) 1 (c) 9 (d) 0
7. 7.What is the smallest number you must add to 508 to make it divisible by 11? (a) 10 (b) 8 (c) 2 (d) 9
8. 8.Which of these numbers is divisible by 11? (a) 3718 (b) 3720 (c) 3717 (d) 3736
9. 9.What is the smallest number you must add to 171 to make it divisible by 7? (a) 5 (b) 4 (c) 6 (d) 3
10. 10.What is the smallest number you must add to 156 to make it divisible by 9? (a) 6 (b) 3 (c) 5 (d) 7
11. 11.What is the units digit of 3^34? (a) 9 (b) 3 (c) 7 (d) 1
12. 12.Which of these numbers is divisible by 4? (a) 1797 (b) 1800 (c) 1801 (d) 1807
13. 13.Which of these numbers is divisible by 11? (a) 7947 (b) 7937 (c) 7931 (d) 7926
14. 14.Which of these numbers is divisible by 3? (a) 2554 (b) 2557 (c) 2552 (d) 2553
15. 15.What is the remainder when 171 is divided by 7? (a) 4 (b) 2 (c) 5 (d) 3
16. 16.What is the remainder when 689 is divided by 12? (a) 6 (b) 4 (c) 7 (d) 5
17. 17.What is the units digit of 3^28? (a) 3 (b) 9 (c) 1 (d) 7
18. 18.What is the smallest number you must add to 152 to make it divisible by 10? (a) 7 (b) 2 (c) 8 (d) 9
19. 19.What is the remainder when 440 is divided by 6? (a) 2 (b) 4 (c) 1 (d) 3
20. 20.What is the units digit of 4^45? (a) 3 (b) 4 (c) 5 (d) 6
21. 21.Which of these numbers is divisible by 9? (a) 3599 (b) 3604 (c) 3610 (d) 3600
22. 22.What is the smallest number you must add to 51 to make it divisible by 4? (a) 3 (b) 1 (c) 2
23. 23.Which of these numbers is divisible by 3? (a) 2344 (b) 2340 (c) 2339 (d) 2342
24. 24.What is the smallest number you must add to 110 to make it divisible by 3? (a) 1 (b) 2
25. 25.Which of these numbers is divisible by 9? (a) 3540 (b) 3552 (c) 3533 (d) 3537
26. 26.What is the units digit of 8^32? (a) 4 (b) 6 (c) 8 (d) 2
27. 27.Which of these numbers is divisible by 6? (a) 4887 (b) 4884 (c) 4883 (d) 4891
28. 28.What is the units digit of 8^36? (a) 6 (b) 8 (c) 2 (d) 4
29. 29.What is the units digit of 7^46? (a) 9 (b) 3 (c) 1 (d) 7
30. 30.What is the smallest number you must add to 88 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
31. 31.What is the smallest number you must add to 184 to make it divisible by 3? (a) 1 (b) 2
32. 32.What is the smallest number you must add to 58 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
33. 33.Which of these numbers is divisible by 8? (a) 2727 (b) 2743 (c) 2728 (d) 2730
34. 34.What is the smallest number you must add to 199 to make it divisible by 9? (a) 6 (b) 8 (c) 7 (d) 1
35. 35.What is the units digit of 9^11? (a) 1 (b) 9 (c) 8 (d) 7
36. 36.Which of these numbers is divisible by 6? (a) 4088 (b) 4095 (c) 4086 (d) 4082
37. 37.Which of these numbers is divisible by 8? (a) 6458 (b) 6443 (c) 6448 (d) 6455
38. 38.What is the remainder when 388 is divided by 11? (a) 4 (b) 3 (c) 8 (d) 2
39. 39.What is the remainder when 214 is divided by 12? (a) 9 (b) 2 (c) 11 (d) 10
40. 40.What is the units digit of 2^13? (a) 4 (b) 8 (c) 6 (d) 2

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