Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 113 · 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 smallest number you must add to 93 to make it divisible by 4? (a) 2 (b) 1 (c) 3
2. 2.What is the remainder when 237 is divided by 9? (a) 2 (b) 6 (c) 3 (d) 4
3. 3.What is the remainder when 38 is divided by 3? (a) 1 (b) 0 (c) 2
4. 4.What is the units digit of 4^20? (a) 5 (b) 6 (c) 4 (d) 7
5. 5.What is the remainder when 341 is divided by 9? (a) 7 (b) 8 (c) 1 (d) 6
6. 6.What is the units digit of 3^31? (a) 3 (b) 1 (c) 9 (d) 7
7. 7.What is the remainder when 523 is divided by 7? (a) 4 (b) 5 (c) 6 (d) 2
8. 8.Which of these numbers is divisible by 9? (a) 8032 (b) 8037 (c) 8048 (d) 8039
9. 9.What is the units digit of 8^38? (a) 4 (b) 8 (c) 6 (d) 2
10. 10.What is the remainder when 79 is divided by 7? (a) 2 (b) 1 (c) 3 (d) 5
11. 11.What is the smallest number you must add to 535 to make it divisible by 7? (a) 5 (b) 3 (c) 4 (d) 6
12. 12.What is the smallest number you must add to 258 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
13. 13.What is the smallest number you must add to 894 to make it divisible by 12? (a) 5 (b) 7 (c) 6 (d) 8
14. 14.What is the remainder when 129 is divided by 7? (a) 2 (b) 4 (c) 3 (d) 5
15. 15.What is the smallest number you must add to 136 to make it divisible by 9? (a) 8 (b) 6 (c) 1 (d) 7
16. 16.What is the units digit of 3^27? (a) 1 (b) 7 (c) 9 (d) 3
17. 17.What is the units digit of 7^26? (a) 1 (b) 3 (c) 7 (d) 9
18. 18.What is the units digit of 4^23? (a) 3 (b) 6 (c) 4 (d) 5
19. 19.What is the units digit of 9^34? (a) 2 (b) 1 (c) 0 (d) 9
20. 20.What is the smallest number you must add to 158 to make it divisible by 8? (a) 3 (b) 2 (c) 1 (d) 6
21. 21.What is the units digit of 7^23? (a) 1 (b) 3 (c) 7 (d) 9
22. 22.Which of these numbers is divisible by 9? (a) 6003 (b) 6009 (c) 6016 (d) 5995
23. 23.Which of these numbers is divisible by 4? (a) 435 (b) 439 (c) 431 (d) 432
24. 24.What is the smallest number you must add to 308 to make it divisible by 8? (a) 5 (b) 6 (c) 4 (d) 3
25. 25.What is the remainder when 361 is divided by 6? (a) 0 (b) 1 (c) 2 (d) 5
26. 26.What is the remainder when 253 is divided by 7? (a) 6 (b) 1 (c) 2 (d) 0
27. 27.What is the smallest number you must add to 249 to make it divisible by 4? (a) 1 (b) 2 (c) 3
28. 28.Which of these numbers is divisible by 3? (a) 1633 (b) 1635 (c) 1639 (d) 1636
29. 29.What is the remainder when 474 is divided by 8? (a) 1 (b) 3 (c) 6 (d) 2
30. 30.What is the units digit of 7^21? (a) 7 (b) 3 (c) 1 (d) 9
31. 31.What is the remainder when 246 is divided by 5? (a) 4 (b) 1 (c) 0 (d) 2
32. 32.What is the smallest number you must add to 185 to make it divisible by 6? (a) 5 (b) 3 (c) 1 (d) 2
33. 33.What is the remainder when 356 is divided by 5? (a) 2 (b) 1 (c) 0 (d) 4
34. 34.Which of these numbers is divisible by 9? (a) 1598 (b) 1602 (c) 1608 (d) 1613
35. 35.Which of these numbers is divisible by 6? (a) 913 (b) 922 (c) 912 (d) 911
36. 36.What is the remainder when 56 is divided by 5? (a) 0 (b) 2 (c) 1 (d) 4
37. 37.What is the remainder when 204 is divided by 9? (a) 5 (b) 7 (c) 6 (d) 3
38. 38.Which of these numbers is divisible by 4? (a) 599 (b) 593 (c) 596 (d) 601
39. 39.What is the smallest number you must add to 783 to make it divisible by 11? (a) 9 (b) 2 (c) 10 (d) 8
40. 40.Which of these numbers is divisible by 6? (a) 1754 (b) 1741 (c) 1747 (d) 1746

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