Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 302 · 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 105 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
2. 2.Which of these numbers is divisible by 3? (a) 1504 (b) 1502 (c) 1499 (d) 1500
3. 3.What is the units digit of 4^25? (a) 5 (b) 4 (c) 3 (d) 6
4. 4.What is the units digit of 9^24? (a) 0 (b) 1 (c) 2 (d) 9
5. 5.Which of these numbers is divisible by 11? (a) 5522 (b) 5538 (c) 5518 (d) 5526
6. 6.What is the remainder when 263 is divided by 9? (a) 2 (b) 3 (c) 1 (d) 7
7. 7.What is the units digit of 8^12? (a) 8 (b) 2 (c) 6 (d) 4
8. 8.What is the smallest number you must add to 868 to make it divisible by 12? (a) 7 (b) 9 (c) 8 (d) 4
9. 9.What is the smallest number you must add to 385 to make it divisible by 8? (a) 5 (b) 1 (c) 7 (d) 6
10. 10.Which of these numbers is divisible by 8? (a) 3408 (b) 3409 (c) 3422 (d) 3404
11. 11.What is the units digit of 8^16? (a) 2 (b) 8 (c) 6 (d) 4
12. 12.What is the smallest number you must add to 271 to make it divisible by 9? (a) 1 (b) 7 (c) 8 (d) 6
13. 13.What is the smallest number you must add to 355 to make it divisible by 8? (a) 4 (b) 5 (c) 6 (d) 3
14. 14.What is the units digit of 9^23? (a) 8 (b) 9 (c) 1 (d) 7
15. 15.What is the smallest number you must add to 547 to make it divisible by 10? (a) 4 (b) 2 (c) 3 (d) 7
16. 16.What is the units digit of 8^26? (a) 8 (b) 4 (c) 6 (d) 2
17. 17.What is the remainder when 890 is divided by 12? (a) 3 (b) 1 (c) 10 (d) 2
18. 18.What is the remainder when 406 is divided by 11? (a) 1 (b) 8 (c) 9 (d) 10
19. 19.What is the units digit of 8^43? (a) 4 (b) 8 (c) 6 (d) 2
20. 20.Which of these numbers is divisible by 3? (a) 138 (b) 140 (c) 136 (d) 142
21. 21.What is the units digit of 9^21? (a) 7 (b) 1 (c) 8 (d) 9
22. 22.What is the remainder when 157 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
23. 23.What is the units digit of 3^53? (a) 7 (b) 1 (c) 3 (d) 9
24. 24.Which of these numbers is divisible by 11? (a) 3978 (b) 3959 (c) 3970 (d) 3960
25. 25.What is the remainder when 221 is divided by 6? (a) 5 (b) 4 (c) 3 (d) 1
26. 26.Which of these numbers is divisible by 4? (a) 1199 (b) 1192 (c) 1191 (d) 1195
27. 27.What is the units digit of 3^60? (a) 1 (b) 7 (c) 3 (d) 9
28. 28.What is the units digit of 2^20? (a) 8 (b) 6 (c) 4 (d) 2
29. 29.What is the remainder when 176 is divided by 7? (a) 2 (b) 1 (c) 6 (d) 0
30. 30.Which of these numbers is divisible by 11? (a) 3513 (b) 3501 (c) 3526 (d) 3509
31. 31.Which of these numbers is divisible by 11? (a) 7260 (b) 7258 (c) 7274 (d) 7263
32. 32.What is the remainder when 111 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 0
33. 33.Which of these numbers is divisible by 6? (a) 834 (b) 838 (c) 830 (d) 845
34. 34.What is the units digit of 7^41? (a) 3 (b) 1 (c) 7 (d) 9
35. 35.What is the smallest number you must add to 613 to make it divisible by 11? (a) 4 (b) 3 (c) 8 (d) 2
36. 36.Which of these numbers is divisible by 11? (a) 4815 (b) 4821 (c) 4801 (d) 4807
37. 37.What is the remainder when 351 is divided by 10? (a) 9 (b) 1 (c) 0 (d) 2
38. 38.Which of these numbers is divisible by 4? (a) 622 (b) 617 (c) 615 (d) 616
39. 39.What is the smallest number you must add to 76 to make it divisible by 3? (a) 1 (b) 2
40. 40.What is the smallest number you must add to 41 to make it divisible by 3? (a) 1 (b) 2

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