Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 642 · 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 82 to make it divisible by 3? (a) 2 (b) 1
2. 2.Which of these numbers is divisible by 6? (a) 3545 (b) 3536 (c) 3534 (d) 3531
3. 3.What is the remainder when 193 is divided by 3? (a) 1 (b) 2 (c) 0
4. 4.Which of these numbers is divisible by 11? (a) 3444 (b) 3425 (c) 3436 (d) 3432
5. 5.Which of these numbers is divisible by 9? (a) 6889 (b) 6876 (c) 6875 (d) 6879
6. 6.What is the remainder when 432 is divided by 10? (a) 8 (b) 1 (c) 3 (d) 2
7. 7.What is the smallest number you must add to 176 to make it divisible by 10? (a) 6 (b) 3 (c) 4 (d) 5
8. 8.What is the smallest number you must add to 253 to make it divisible by 4? (a) 1 (b) 3 (c) 2
9. 9.What is the smallest number you must add to 377 to make it divisible by 6? (a) 2 (b) 1 (c) 3 (d) 5
10. 10.Which of these numbers is divisible by 6? (a) 1002 (b) 997 (c) 1013 (d) 1006
11. 11.What is the remainder when 204 is divided by 9? (a) 7 (b) 3 (c) 5 (d) 6
12. 12.What is the units digit of 2^14? (a) 4 (b) 6 (c) 8 (d) 2
13. 13.What is the units digit of 8^22? (a) 6 (b) 4 (c) 8 (d) 2
14. 14.What is the units digit of 8^16? (a) 8 (b) 6 (c) 4 (d) 2
15. 15.Which of these numbers is divisible by 4? (a) 1368 (b) 1373 (c) 1367 (d) 1369
16. 16.What is the units digit of 8^52? (a) 8 (b) 6 (c) 2 (d) 4
17. 17.Which of these numbers is divisible by 11? (a) 4928 (b) 4945 (c) 4929 (d) 4924
18. 18.What is the smallest number you must add to 217 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
19. 19.Which of these numbers is divisible by 4? (a) 2229 (b) 2226 (c) 2224 (d) 2221
20. 20.What is the remainder when 612 is divided by 10? (a) 2 (b) 1 (c) 3 (d) 8
21. 21.What is the smallest number you must add to 244 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 1
22. 22.Which of these numbers is divisible by 4? (a) 3170 (b) 3167 (c) 3164 (d) 3163
23. 23.Which of these numbers is divisible by 3? (a) 813 (b) 814 (c) 818 (d) 812
24. 24.What is the smallest number you must add to 32 to make it divisible by 3? (a) 1 (b) 2
25. 25.Which of these numbers is divisible by 4? (a) 1075 (b) 1077 (c) 1069 (d) 1072
26. 26.Which of these numbers is divisible by 4? (a) 3114 (b) 3108 (c) 3106 (d) 3109
27. 27.What is the units digit of 2^30? (a) 6 (b) 2 (c) 8 (d) 4
28. 28.What is the smallest number you must add to 202 to make it divisible by 9? (a) 7 (b) 5 (c) 4 (d) 6
29. 29.What is the remainder when 298 is divided by 9? (a) 2 (b) 1 (c) 8 (d) 0
30. 30.Which of these numbers is divisible by 4? (a) 2698 (b) 2701 (c) 2693 (d) 2696
31. 31.What is the smallest number you must add to 272 to make it divisible by 6? (a) 4 (b) 5 (c) 2 (d) 3
32. 32.What is the remainder when 741 is divided by 10? (a) 0 (b) 1 (c) 2 (d) 9
33. 33.What is the remainder when 44 is divided by 3? (a) 0 (b) 1 (c) 2
34. 34.Which of these numbers is divisible by 9? (a) 4516 (b) 4532 (c) 4520 (d) 4518
35. 35.What is the remainder when 127 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
36. 36.Which of these numbers is divisible by 9? (a) 7981 (b) 7999 (c) 7991 (d) 7983
37. 37.Which of these numbers is divisible by 11? (a) 2066 (b) 2070 (c) 2055 (d) 2057
38. 38.What is the remainder when 697 is divided by 12? (a) 2 (b) 0 (c) 11 (d) 1
39. 39.What is the units digit of 2^28? (a) 8 (b) 2 (c) 4 (d) 6
40. 40.What is the remainder when 509 is divided by 7? (a) 2 (b) 4 (c) 5 (d) 6

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