Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 959 · 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 889 to make it divisible by 12? (a) 11 (b) 10 (c) 1 (d) 9
2. 2.What is the smallest number you must add to 137 to make it divisible by 11? (a) 8 (b) 5 (c) 7 (d) 6
3. 3.Which of these numbers is divisible by 4? (a) 3306 (b) 3309 (c) 3313 (d) 3308
4. 4.What is the remainder when 199 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
5. 5.What is the smallest number you must add to 209 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
6. 6.Which of these numbers is divisible by 4? (a) 809 (b) 807 (c) 814 (d) 808
7. 7.Which of these numbers is divisible by 9? (a) 783 (b) 788 (c) 775 (d) 795
8. 8.What is the units digit of 9^15? (a) 9 (b) 8 (c) 7 (d) 1
9. 9.What is the units digit of 4^36? (a) 6 (b) 7 (c) 4 (d) 5
10. 10.Which of these numbers is divisible by 4? (a) 2281 (b) 2287 (c) 2280 (d) 2279
11. 11.What is the smallest number you must add to 401 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 2
12. 12.Which of these numbers is divisible by 8? (a) 1589 (b) 1570 (c) 1576 (d) 1580
13. 13.Which of these numbers is divisible by 4? (a) 972 (b) 971 (c) 978 (d) 973
14. 14.Which of these numbers is divisible by 8? (a) 6552 (b) 6561 (c) 6558 (d) 6547
15. 15.Which of these numbers is divisible by 4? (a) 3131 (b) 3132 (c) 3134 (d) 3138
16. 16.Which of these numbers is divisible by 11? (a) 9852 (b) 9845 (c) 9836 (d) 9860
17. 17.What is the smallest number you must add to 91 to make it divisible by 3? (a) 1 (b) 2
18. 18.What is the units digit of 8^47? (a) 4 (b) 8 (c) 6 (d) 2
19. 19.Which of these numbers is divisible by 9? (a) 891 (b) 894 (c) 887 (d) 901
20. 20.Which of these numbers is divisible by 6? (a) 1273 (b) 1271 (c) 1272 (d) 1281
21. 21.Which of these numbers is divisible by 6? (a) 4885 (b) 4890 (c) 4901 (d) 4894
22. 22.What is the units digit of 3^40? (a) 3 (b) 7 (c) 9 (d) 1
23. 23.What is the remainder when 169 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
24. 24.What is the remainder when 230 is divided by 6? (a) 4 (b) 3 (c) 2 (d) 1
25. 25.What is the units digit of 2^23? (a) 8 (b) 4 (c) 2 (d) 6
26. 26.Which of these numbers is divisible by 9? (a) 6375 (b) 6363 (c) 6366 (d) 6357
27. 27.What is the smallest number you must add to 601 to make it divisible by 10? (a) 8 (b) 1 (c) 9 (d) 7
28. 28.Which of these numbers is divisible by 9? (a) 3548 (b) 3546 (c) 3556 (d) 3538
29. 29.What is the units digit of 9^22? (a) 9 (b) 1 (c) 0 (d) 2
30. 30.What is the units digit of 4^13? (a) 6 (b) 3 (c) 5 (d) 4
31. 31.What is the units digit of 9^21? (a) 8 (b) 7 (c) 9 (d) 1
32. 32.What is the smallest number you must add to 346 to make it divisible by 6? (a) 1 (b) 4 (c) 3 (d) 2
33. 33.What is the units digit of 4^11? (a) 3 (b) 4 (c) 5 (d) 6
34. 34.What is the smallest number you must add to 478 to make it divisible by 10? (a) 3 (b) 1 (c) 8 (d) 2
35. 35.Which of these numbers is divisible by 8? (a) 4665 (b) 4653 (c) 4661 (d) 4656
36. 36.Which of these numbers is divisible by 8? (a) 5836 (b) 5824 (c) 5822 (d) 5826
37. 37.What is the smallest number you must add to 468 to make it divisible by 8? (a) 5 (b) 3 (c) 6 (d) 4
38. 38.What is the remainder when 311 is divided by 5? (a) 4 (b) 0 (c) 2 (d) 1
39. 39.Which of these numbers is divisible by 9? (a) 3535 (b) 3537 (c) 3540 (d) 3550
40. 40.What is the remainder when 46 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1

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