Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 464 · 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 65 to make it divisible by 6? (a) 1 (b) 2 (c) 3 (d) 5
2. 2.Which of these numbers is divisible by 9? (a) 1620 (b) 1616 (c) 1621 (d) 1631
3. 3.Which of these numbers is divisible by 3? (a) 1103 (b) 1100 (c) 1098 (d) 1097
4. 4.What is the remainder when 183 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
5. 5.What is the units digit of 8^23? (a) 4 (b) 6 (c) 8 (d) 2
6. 6.What is the units digit of 2^31? (a) 2 (b) 8 (c) 4 (d) 6
7. 7.What is the smallest number you must add to 229 to make it divisible by 7? (a) 5 (b) 1 (c) 2 (d) 3
8. 8.What is the units digit of 8^54? (a) 2 (b) 6 (c) 4 (d) 8
9. 9.What is the smallest number you must add to 491 to make it divisible by 7? (a) 6 (b) 4 (c) 1 (d) 5
10. 10.What is the units digit of 9^23? (a) 7 (b) 9 (c) 1 (d) 8
11. 11.What is the units digit of 7^37? (a) 7 (b) 1 (c) 9 (d) 3
12. 12.What is the smallest number you must add to 152 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
13. 13.Which of these numbers is divisible by 4? (a) 3024 (b) 3023 (c) 3025 (d) 3029
14. 14.What is the units digit of 2^40? (a) 6 (b) 4 (c) 2 (d) 8
15. 15.What is the remainder when 532 is divided by 8? (a) 6 (b) 5 (c) 4 (d) 3
16. 16.What is the smallest number you must add to 437 to make it divisible by 6? (a) 3 (b) 1 (c) 2 (d) 5
17. 17.Which of these numbers is divisible by 4? (a) 938 (b) 943 (c) 936 (d) 933
18. 18.What is the units digit of 2^50? (a) 4 (b) 2 (c) 8 (d) 6
19. 19.What is the smallest number you must add to 226 to make it divisible by 3? (a) 2 (b) 1
20. 20.What is the smallest number you must add to 453 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 4
21. 21.Which of these numbers is divisible by 4? (a) 2152 (b) 2157 (c) 2153 (d) 2149
22. 22.Which of these numbers is divisible by 3? (a) 2041 (b) 2038 (c) 2037 (d) 2036
23. 23.What is the units digit of 4^11? (a) 3 (b) 4 (c) 5 (d) 6
24. 24.What is the units digit of 2^18? (a) 2 (b) 4 (c) 8 (d) 6
25. 25.What is the remainder when 107 is divided by 3? (a) 2 (b) 1 (c) 0
26. 26.What is the remainder when 637 is divided by 12? (a) 1 (b) 0 (c) 2 (d) 11
27. 27.What is the units digit of 2^46? (a) 4 (b) 6 (c) 2 (d) 8
28. 28.What is the smallest number you must add to 203 to make it divisible by 11? (a) 8 (b) 5 (c) 7 (d) 6
29. 29.What is the remainder when 179 is divided by 3? (a) 0 (b) 2 (c) 1
30. 30.Which of these numbers is divisible by 6? (a) 3684 (b) 3692 (c) 3686 (d) 3682
31. 31.What is the remainder when 64 is divided by 3? (a) 2 (b) 1 (c) 0
32. 32.What is the remainder when 521 is divided by 11? (a) 5 (b) 3 (c) 7 (d) 4
33. 33.What is the units digit of 3^47? (a) 9 (b) 3 (c) 7 (d) 1
34. 34.Which of these numbers is divisible by 3? (a) 739 (b) 736 (c) 742 (d) 738
35. 35.Which of these numbers is divisible by 4? (a) 2412 (b) 2418 (c) 2411 (d) 2415
36. 36.What is the units digit of 7^33? (a) 3 (b) 9 (c) 7 (d) 1
37. 37.What is the smallest number you must add to 379 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
38. 38.What is the remainder when 197 is divided by 4? (a) 1 (b) 0 (c) 3 (d) 2
39. 39.What is the units digit of 9^24? (a) 9 (b) 2 (c) 1 (d) 0
40. 40.What is the remainder when 320 is divided by 6? (a) 4 (b) 1 (c) 2 (d) 3

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