Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 454 · 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 107 is divided by 6? (a) 3 (b) 5 (c) 1 (d) 4
2. 2.What is the units digit of 9^44? (a) 1 (b) 0 (c) 2 (d) 9
3. 3.What is the remainder when 265 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
4. 4.What is the remainder when 374 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
5. 5.What is the smallest number you must add to 229 to make it divisible by 3? (a) 1 (b) 2
6. 6.What is the units digit of 8^41? (a) 2 (b) 8 (c) 4 (d) 6
7. 7.What is the smallest number you must add to 137 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
8. 8.What is the units digit of 4^59? (a) 6 (b) 5 (c) 3 (d) 4
9. 9.Which of these numbers is divisible by 6? (a) 998 (b) 1010 (c) 1007 (d) 1002
10. 10.Which of these numbers is divisible by 3? (a) 1358 (b) 1364 (c) 1361 (d) 1359
11. 11.What is the remainder when 307 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
12. 12.Which of these numbers is divisible by 3? (a) 1547 (b) 1550 (c) 1545 (d) 1543
13. 13.What is the smallest number you must add to 221 to make it divisible by 4? (a) 3 (b) 2 (c) 1
14. 14.What is the remainder when 95 is divided by 6? (a) 1 (b) 4 (c) 5 (d) 3
15. 15.Which of these numbers is divisible by 6? (a) 3146 (b) 3154 (c) 3140 (d) 3144
16. 16.What is the remainder when 336 is divided by 10? (a) 6 (b) 5 (c) 7 (d) 4
17. 17.What is the units digit of 8^24? (a) 2 (b) 6 (c) 4 (d) 8
18. 18.Which of these numbers is divisible by 11? (a) 6218 (b) 6213 (c) 6236 (d) 6215
19. 19.Which of these numbers is divisible by 3? (a) 1070 (b) 1075 (c) 1072 (d) 1071
20. 20.What is the remainder when 94 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
21. 21.What is the remainder when 290 is divided by 9? (a) 2 (b) 7 (c) 3 (d) 1
22. 22.What is the units digit of 8^16? (a) 2 (b) 8 (c) 4 (d) 6
23. 23.What is the smallest number you must add to 352 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
24. 24.What is the remainder when 332 is divided by 10? (a) 3 (b) 8 (c) 1 (d) 2
25. 25.Which of these numbers is divisible by 3? (a) 1550 (b) 1552 (c) 1551 (d) 1556
26. 26.Which of these numbers is divisible by 6? (a) 359 (b) 370 (c) 360 (d) 362
27. 27.What is the smallest number you must add to 762 to make it divisible by 10? (a) 9 (b) 7 (c) 2 (d) 8
28. 28.What is the remainder when 198 is divided by 12? (a) 6 (b) 8 (c) 7 (d) 5
29. 29.What is the units digit of 7^56? (a) 9 (b) 7 (c) 3 (d) 1
30. 30.What is the smallest number you must add to 185 to make it divisible by 9? (a) 5 (b) 6 (c) 4 (d) 3
31. 31.Which of these numbers is divisible by 8? (a) 6433 (b) 6430 (c) 6432 (d) 6442
32. 32.Which of these numbers is divisible by 9? (a) 1179 (b) 1189 (c) 1178 (d) 1184
33. 33.What is the remainder when 754 is divided by 10? (a) 6 (b) 4 (c) 5 (d) 3
34. 34.What is the remainder when 586 is divided by 8? (a) 3 (b) 1 (c) 6 (d) 2
35. 35.What is the remainder when 366 is divided by 12? (a) 6 (b) 8 (c) 5 (d) 7
36. 36.What is the remainder when 183 is divided by 12? (a) 3 (b) 4 (c) 2 (d) 9
37. 37.What is the smallest number you must add to 530 to make it divisible by 7? (a) 2 (b) 5 (c) 1 (d) 3
38. 38.Which of these numbers is divisible by 4? (a) 794 (b) 790 (c) 788 (d) 787
39. 39.What is the units digit of 8^31? (a) 4 (b) 8 (c) 2 (d) 6
40. 40.What is the smallest number you must add to 132 to make it divisible by 7? (a) 3 (b) 1 (c) 6 (d) 2

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