Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 506 · 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 11? (a) 8294 (b) 8291 (c) 8307 (d) 8302
2. 2.Which of these numbers is divisible by 9? (a) 554 (b) 549 (c) 560 (d) 546
3. 3.Which of these numbers is divisible by 6? (a) 1186 (b) 1199 (c) 1188 (d) 1191
4. 4.Which of these numbers is divisible by 9? (a) 1061 (b) 1046 (c) 1044 (d) 1039
5. 5.What is the units digit of 4^34? (a) 7 (b) 4 (c) 5 (d) 6
6. 6.What is the remainder when 165 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
7. 7.Which of these numbers is divisible by 8? (a) 4929 (b) 4926 (c) 4928 (d) 4942
8. 8.What is the smallest number you must add to 242 to make it divisible by 7? (a) 3 (b) 2 (c) 5 (d) 4
9. 9.Which of these numbers is divisible by 8? (a) 6427 (b) 6412 (c) 6417 (d) 6416
10. 10.What is the units digit of 8^16? (a) 2 (b) 8 (c) 6 (d) 4
11. 11.What is the units digit of 9^51? (a) 9 (b) 7 (c) 1 (d) 8
12. 12.What is the smallest number you must add to 127 to make it divisible by 3? (a) 1 (b) 2
13. 13.What is the remainder when 488 is divided by 9? (a) 7 (b) 2 (c) 1 (d) 3
14. 14.What is the units digit of 8^13? (a) 8 (b) 4 (c) 6 (d) 2
15. 15.Which of these numbers is divisible by 4? (a) 422 (b) 427 (c) 424 (d) 429
16. 16.Which of these numbers is divisible by 6? (a) 969 (b) 960 (c) 955 (d) 964
17. 17.What is the units digit of 9^58? (a) 0 (b) 1 (c) 9 (d) 2
18. 18.Which of these numbers is divisible by 4? (a) 2160 (b) 2163 (c) 2166 (d) 2159
19. 19.What is the remainder when 70 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
20. 20.What is the remainder when 256 is divided by 5? (a) 2 (b) 0 (c) 4 (d) 1
21. 21.Which of these numbers is divisible by 8? (a) 1873 (b) 1864 (c) 1870 (d) 1857
22. 22.Which of these numbers is divisible by 6? (a) 1813 (b) 1806 (c) 1808 (d) 1804
23. 23.What is the units digit of 4^46? (a) 6 (b) 7 (c) 5 (d) 4
24. 24.What is the remainder when 94 is divided by 3? (a) 0 (b) 1 (c) 2
25. 25.What is the smallest number you must add to 829 to make it divisible by 12? (a) 11 (b) 1 (c) 10 (d) 9
26. 26.Which of these numbers is divisible by 4? (a) 3591 (b) 3593 (c) 3597 (d) 3592
27. 27.Which of these numbers is divisible by 3? (a) 1229 (b) 1232 (c) 1226 (d) 1227
28. 28.Which of these numbers is divisible by 3? (a) 803 (b) 804 (c) 805 (d) 808
29. 29.Which of these numbers is divisible by 9? (a) 846 (b) 843 (c) 860 (d) 849
30. 30.Which of these numbers is divisible by 9? (a) 4331 (b) 4346 (c) 4354 (d) 4338
31. 31.Which of these numbers is divisible by 4? (a) 2547 (b) 2544 (c) 2549 (d) 2543
32. 32.What is the smallest number you must add to 158 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
33. 33.What is the units digit of 4^36? (a) 4 (b) 7 (c) 5 (d) 6
34. 34.What is the remainder when 501 is divided by 9? (a) 3 (b) 5 (c) 7 (d) 6
35. 35.Which of these numbers is divisible by 6? (a) 4386 (b) 4395 (c) 4391 (d) 4382
36. 36.Which of these numbers is divisible by 9? (a) 4914 (b) 4906 (c) 4924 (d) 4916
37. 37.Which of these numbers is divisible by 8? (a) 802 (b) 798 (c) 813 (d) 800
38. 38.Which of these numbers is divisible by 6? (a) 2882 (b) 2875 (c) 2890 (d) 2880
39. 39.Which of these numbers is divisible by 11? (a) 6727 (b) 6734 (c) 6732 (d) 6752
40. 40.What is the smallest number you must add to 175 to make it divisible by 11? (a) 3 (b) 2 (c) 1 (d) 10

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