Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 462 · 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 units digit of 2^52? (a) 4 (b) 6 (c) 2 (d) 8
2. 2.What is the units digit of 4^23? (a) 3 (b) 6 (c) 5 (d) 4
3. 3.Which of these numbers is divisible by 8? (a) 6582 (b) 6561 (c) 6575 (d) 6568
4. 4.Which of these numbers is divisible by 4? (a) 3396 (b) 3402 (c) 3393 (d) 3397
5. 5.What is the units digit of 2^37? (a) 6 (b) 4 (c) 2 (d) 8
6. 6.What is the remainder when 166 is divided by 6? (a) 3 (b) 2 (c) 4 (d) 5
7. 7.What is the units digit of 3^47? (a) 7 (b) 3 (c) 9 (d) 1
8. 8.What is the smallest number you must add to 418 to make it divisible by 6? (a) 4 (b) 1 (c) 2 (d) 3
9. 9.What is the remainder when 143 is divided by 3? (a) 1 (b) 0 (c) 2
10. 10.What is the remainder when 712 is divided by 12? (a) 5 (b) 8 (c) 3 (d) 4
11. 11.What is the remainder when 311 is divided by 5? (a) 2 (b) 0 (c) 4 (d) 1
12. 12.What is the smallest number you must add to 479 to make it divisible by 6? (a) 1 (b) 5 (c) 3 (d) 2
13. 13.What is the units digit of 8^14? (a) 6 (b) 2 (c) 4 (d) 8
14. 14.What is the remainder when 512 is divided by 10? (a) 1 (b) 8 (c) 2 (d) 3
15. 15.What is the units digit of 9^35? (a) 7 (b) 9 (c) 8 (d) 1
16. 16.What is the remainder when 241 is divided by 7? (a) 4 (b) 3 (c) 2 (d) 5
17. 17.Which of these numbers is divisible by 3? (a) 334 (b) 329 (c) 330 (d) 331
18. 18.What is the units digit of 9^60? (a) 1 (b) 2 (c) 9 (d) 0
19. 19.What is the smallest number you must add to 599 to make it divisible by 9? (a) 5 (b) 6 (c) 4 (d) 3
20. 20.What is the units digit of 8^31? (a) 2 (b) 4 (c) 6 (d) 8
21. 21.What is the remainder when 243 is divided by 11? (a) 2 (b) 0 (c) 1 (d) 10
22. 22.Which of these numbers is divisible by 9? (a) 3483 (b) 3489 (c) 3476 (d) 3493
23. 23.What is the units digit of 4^54? (a) 6 (b) 4 (c) 5 (d) 7
24. 24.Which of these numbers is divisible by 4? (a) 2753 (b) 2751 (c) 2758 (d) 2752
25. 25.What is the remainder when 148 is divided by 3? (a) 2 (b) 0 (c) 1
26. 26.What is the smallest number you must add to 166 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
27. 27.What is the units digit of 9^49? (a) 9 (b) 8 (c) 1 (d) 7
28. 28.Which of these numbers is divisible by 4? (a) 2656 (b) 2653 (c) 2658 (d) 2661
29. 29.What is the remainder when 142 is divided by 3? (a) 1 (b) 0 (c) 2
30. 30.What is the units digit of 7^17? (a) 3 (b) 9 (c) 1 (d) 7
31. 31.What is the smallest number you must add to 434 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 2
32. 32.What is the remainder when 339 is divided by 8? (a) 5 (b) 4 (c) 2 (d) 3
33. 33.What is the smallest number you must add to 789 to make it divisible by 12? (a) 2 (b) 9 (c) 3 (d) 4
34. 34.What is the remainder when 157 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
35. 35.What is the smallest number you must add to 663 to make it divisible by 9? (a) 2 (b) 3 (c) 4 (d) 6
36. 36.What is the units digit of 9^52? (a) 9 (b) 2 (c) 0 (d) 1
37. 37.Which of these numbers is divisible by 9? (a) 3200 (b) 3207 (c) 3195 (d) 3188
38. 38.What is the units digit of 8^35? (a) 8 (b) 4 (c) 2 (d) 6
39. 39.What is the remainder when 124 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
40. 40.What is the smallest number you must add to 81 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 4

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