Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 841 · 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 743 is divided by 10? (a) 3 (b) 7 (c) 2 (d) 4
2. 2.What is the units digit of 7^10? (a) 1 (b) 9 (c) 3 (d) 7
3. 3.Which of these numbers is divisible by 3? (a) 484 (b) 482 (c) 478 (d) 480
4. 4.Which of these numbers is divisible by 8? (a) 4144 (b) 4150 (c) 4137 (d) 4154
5. 5.What is the remainder when 606 is divided by 9? (a) 4 (b) 6 (c) 2 (d) 3
6. 6.What is the smallest number you must add to 422 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 4
7. 7.Which of these numbers is divisible by 6? (a) 2832 (b) 2843 (c) 2828 (d) 2835
8. 8.What is the remainder when 293 is divided by 12? (a) 7 (b) 4 (c) 5 (d) 6
9. 9.What is the units digit of 3^32? (a) 3 (b) 9 (c) 1 (d) 7
10. 10.What is the remainder when 793 is divided by 10? (a) 3 (b) 2 (c) 4 (d) 7
11. 11.What is the remainder when 152 is divided by 6? (a) 1 (b) 4 (c) 2 (d) 3
12. 12.Which of these numbers is divisible by 11? (a) 6936 (b) 6962 (c) 6941 (d) 6944
13. 13.What is the remainder when 159 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
14. 14.What is the smallest number you must add to 74 to make it divisible by 3? (a) 1 (b) 2
15. 15.What is the units digit of 8^54? (a) 6 (b) 2 (c) 4 (d) 8
16. 16.What is the smallest number you must add to 169 to make it divisible by 3? (a) 1 (b) 2
17. 17.What is the smallest number you must add to 361 to make it divisible by 9? (a) 1 (b) 7 (c) 8 (d) 6
18. 18.What is the smallest number you must add to 153 to make it divisible by 11? (a) 3 (b) 10 (c) 2 (d) 1
19. 19.Which of these numbers is divisible by 9? (a) 1648 (b) 1646 (c) 1647 (d) 1658
20. 20.What is the units digit of 2^45? (a) 2 (b) 4 (c) 6 (d) 8
21. 21.What is the smallest number you must add to 501 to make it divisible by 12? (a) 9 (b) 2 (c) 4 (d) 3
22. 22.What is the units digit of 2^57? (a) 4 (b) 6 (c) 8 (d) 2
23. 23.Which of these numbers is divisible by 3? (a) 2463 (b) 2467 (c) 2462 (d) 2464
24. 24.What is the units digit of 9^51? (a) 1 (b) 7 (c) 9 (d) 8
25. 25.What is the remainder when 218 is divided by 3? (a) 1 (b) 2 (c) 0
26. 26.Which of these numbers is divisible by 4? (a) 1487 (b) 1482 (c) 1480 (d) 1478
27. 27.Which of these numbers is divisible by 8? (a) 1700 (b) 1694 (c) 1681 (d) 1688
28. 28.What is the units digit of 3^35? (a) 9 (b) 3 (c) 7 (d) 1
29. 29.What is the units digit of 9^33? (a) 7 (b) 8 (c) 1 (d) 9
30. 30.What is the units digit of 8^48? (a) 4 (b) 8 (c) 6 (d) 2
31. 31.What is the units digit of 8^36? (a) 8 (b) 2 (c) 4 (d) 6
32. 32.What is the remainder when 86 is divided by 3? (a) 2 (b) 0 (c) 1
33. 33.Which of these numbers is divisible by 11? (a) 1971 (b) 1980 (c) 1990 (d) 1997
34. 34.What is the remainder when 499 is divided by 11? (a) 3 (b) 4 (c) 7 (d) 5
35. 35.What is the remainder when 481 is divided by 6? (a) 5 (b) 0 (c) 2 (d) 1
36. 36.What is the remainder when 122 is divided by 6? (a) 4 (b) 2 (c) 1 (d) 3
37. 37.What is the smallest number you must add to 164 to make it divisible by 9? (a) 8 (b) 6 (c) 7 (d) 2
38. 38.What is the remainder when 114 is divided by 9? (a) 3 (b) 6 (c) 7 (d) 5
39. 39.What is the smallest number you must add to 738 to make it divisible by 12? (a) 7 (b) 8 (c) 5 (d) 6
40. 40.What is the remainder when 63 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1

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