Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 152 · 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 306 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 0
2. 2.What is the units digit of 9^45? (a) 1 (b) 7 (c) 9 (d) 8
3. 3.Which of these numbers is divisible by 9? (a) 7686 (b) 7682 (c) 7692 (d) 7699
4. 4.What is the smallest number you must add to 169 to make it divisible by 7? (a) 1 (b) 4 (c) 6 (d) 5
5. 5.What is the smallest number you must add to 331 to make it divisible by 8? (a) 4 (b) 6 (c) 5 (d) 3
6. 6.What is the smallest number you must add to 124 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
7. 7.What is the smallest number you must add to 118 to make it divisible by 6? (a) 1 (b) 2 (c) 4 (d) 3
8. 8.What is the units digit of 8^40? (a) 6 (b) 2 (c) 8 (d) 4
9. 9.Which of these numbers is divisible by 4? (a) 1283 (b) 1274 (c) 1279 (d) 1276
10. 10.What is the units digit of 8^32? (a) 6 (b) 2 (c) 8 (d) 4
11. 11.What is the units digit of 3^52? (a) 3 (b) 7 (c) 9 (d) 1
12. 12.What is the remainder when 153 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
13. 13.What is the units digit of 4^29? (a) 3 (b) 4 (c) 6 (d) 5
14. 14.Which of these numbers is divisible by 3? (a) 824 (b) 820 (c) 827 (d) 822
15. 15.What is the smallest number you must add to 624 to make it divisible by 10? (a) 7 (b) 4 (c) 5 (d) 6
16. 16.What is the units digit of 7^60? (a) 3 (b) 7 (c) 1 (d) 9
17. 17.What is the units digit of 9^42? (a) 9 (b) 0 (c) 2 (d) 1
18. 18.Which of these numbers is divisible by 8? (a) 3043 (b) 3025 (c) 3032 (d) 3039
19. 19.Which of these numbers is divisible by 11? (a) 8635 (b) 8636 (c) 8656 (d) 8628
20. 20.Which of these numbers is divisible by 8? (a) 3591 (b) 3584 (c) 3581 (d) 3594
21. 21.What is the smallest number you must add to 846 to make it divisible by 12? (a) 5 (b) 6 (c) 7 (d) 8
22. 22.What is the remainder when 213 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
23. 23.What is the smallest number you must add to 173 to make it divisible by 3? (a) 2 (b) 1
24. 24.What is the remainder when 710 is divided by 9? (a) 1 (b) 8 (c) 6 (d) 7
25. 25.Which of these numbers is divisible by 3? (a) 2351 (b) 2352 (c) 2353 (d) 2357
26. 26.What is the units digit of 4^21? (a) 3 (b) 6 (c) 4 (d) 5
27. 27.What is the units digit of 9^14? (a) 1 (b) 2 (c) 9 (d) 0
28. 28.What is the smallest number you must add to 43 to make it divisible by 3? (a) 1 (b) 2
29. 29.What is the units digit of 9^17? (a) 8 (b) 9 (c) 7 (d) 1
30. 30.What is the remainder when 668 is divided by 12? (a) 9 (b) 7 (c) 4 (d) 8
31. 31.What is the units digit of 2^38? (a) 4 (b) 6 (c) 8 (d) 2
32. 32.What is the remainder when 258 is divided by 8? (a) 6 (b) 1 (c) 2 (d) 3
33. 33.What is the remainder when 423 is divided by 11? (a) 5 (b) 6 (c) 7 (d) 4
34. 34.What is the units digit of 9^27? (a) 1 (b) 7 (c) 9 (d) 8
35. 35.What is the smallest number you must add to 119 to make it divisible by 3? (a) 1 (b) 2
36. 36.Which of these numbers is divisible by 11? (a) 1782 (b) 1783 (c) 1798 (d) 1775
37. 37.What is the smallest number you must add to 76 to make it divisible by 3? (a) 2 (b) 1
38. 38.Which of these numbers is divisible by 11? (a) 3436 (b) 3432 (c) 3422 (d) 3449
39. 39.What is the units digit of 8^37? (a) 8 (b) 4 (c) 6 (d) 2
40. 40.What is the remainder when 689 is divided by 9? (a) 5 (b) 6 (c) 7 (d) 4

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