Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 91 · 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 2^32? (a) 6 (b) 2 (c) 8 (d) 4
2. 2.What is the remainder when 357 is divided by 8? (a) 3 (b) 4 (c) 5 (d) 6
3. 3.What is the smallest number you must add to 832 to make it divisible by 12? (a) 4 (b) 8 (c) 9 (d) 7
4. 4.Which of these numbers is divisible by 4? (a) 587 (b) 582 (c) 584 (d) 589
5. 5.Which of these numbers is divisible by 4? (a) 3236 (b) 3239 (c) 3241 (d) 3233
6. 6.What is the remainder when 630 is divided by 8? (a) 5 (b) 2 (c) 6 (d) 7
7. 7.What is the remainder when 793 is divided by 10? (a) 7 (b) 2 (c) 3 (d) 4
8. 8.What is the units digit of 7^43? (a) 1 (b) 9 (c) 7 (d) 3
9. 9.What is the smallest number you must add to 166 to make it divisible by 7? (a) 2 (b) 1 (c) 5 (d) 3
10. 10.Which of these numbers is divisible by 6? (a) 5330 (b) 5328 (c) 5324 (d) 5335
11. 11.What is the smallest number you must add to 406 to make it divisible by 6? (a) 4 (b) 2 (c) 1 (d) 3
12. 12.Which of these numbers is divisible by 6? (a) 2780 (b) 2778 (c) 2775 (d) 2789
13. 13.What is the remainder when 93 is divided by 9? (a) 3 (b) 4 (c) 6 (d) 2
14. 14.What is the units digit of 7^16? (a) 3 (b) 7 (c) 1 (d) 9
15. 15.Which of these numbers is divisible by 11? (a) 6187 (b) 6198 (c) 6182 (d) 6176
16. 16.Which of these numbers is divisible by 3? (a) 2646 (b) 2645 (c) 2650 (d) 2647
17. 17.What is the units digit of 2^26? (a) 8 (b) 4 (c) 2 (d) 6
18. 18.Which of these numbers is divisible by 8? (a) 1171 (b) 1190 (c) 1180 (d) 1176
19. 19.What is the remainder when 62 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
20. 20.What is the remainder when 59 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
21. 21.Which of these numbers is divisible by 4? (a) 3561 (b) 3555 (c) 3556 (d) 3557
22. 22.Which of these numbers is divisible by 11? (a) 2509 (b) 2488 (c) 2497 (d) 2498
23. 23.What is the units digit of 2^13? (a) 2 (b) 6 (c) 8 (d) 4
24. 24.What is the units digit of 9^32? (a) 2 (b) 0 (c) 9 (d) 1
25. 25.What is the units digit of 3^48? (a) 7 (b) 1 (c) 9 (d) 3
26. 26.What is the remainder when 286 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 0
27. 27.What is the smallest number you must add to 267 to make it divisible by 4? (a) 2 (b) 1 (c) 3
28. 28.What is the remainder when 545 is divided by 7? (a) 1 (b) 5 (c) 4 (d) 6
29. 29.What is the remainder when 264 is divided by 9? (a) 2 (b) 4 (c) 3 (d) 6
30. 30.What is the remainder when 399 is divided by 9? (a) 3 (b) 4 (c) 2 (d) 6
31. 31.What is the remainder when 467 is divided by 12? (a) 10 (b) 1 (c) 9 (d) 11
32. 32.What is the units digit of 4^21? (a) 3 (b) 6 (c) 4 (d) 5
33. 33.What is the smallest number you must add to 731 to make it divisible by 11? (a) 8 (b) 7 (c) 6 (d) 5
34. 34.What is the smallest number you must add to 317 to make it divisible by 4? (a) 1 (b) 2 (c) 3
35. 35.What is the units digit of 7^57? (a) 7 (b) 9 (c) 1 (d) 3
36. 36.What is the remainder when 77 is divided by 3? (a) 0 (b) 2 (c) 1
37. 37.What is the remainder when 148 is divided by 7? (a) 2 (b) 6 (c) 0 (d) 1
38. 38.What is the remainder when 72 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
39. 39.What is the units digit of 7^17? (a) 7 (b) 9 (c) 3 (d) 1
40. 40.What is the units digit of 3^21? (a) 7 (b) 1 (c) 3 (d) 9

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