Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 499 · 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 3^24? (a) 9 (b) 3 (c) 1 (d) 7
2. 2.Which of these numbers is divisible by 4? (a) 528 (b) 529 (c) 533 (d) 527
3. 3.What is the smallest number you must add to 178 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
4. 4.What is the remainder when 107 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
5. 5.What is the remainder when 90 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
6. 6.What is the remainder when 125 is divided by 3? (a) 2 (b) 1 (c) 0
7. 7.What is the units digit of 4^51? (a) 3 (b) 4 (c) 6 (d) 5
8. 8.Which of these numbers is divisible by 9? (a) 6315 (b) 6320 (c) 6330 (d) 6318
9. 9.What is the remainder when 175 is divided by 3? (a) 2 (b) 0 (c) 1
10. 10.What is the smallest number you must add to 136 to make it divisible by 3? (a) 2 (b) 1
11. 11.What is the smallest number you must add to 178 to make it divisible by 4? (a) 3 (b) 2 (c) 1
12. 12.What is the remainder when 137 is divided by 9? (a) 2 (b) 3 (c) 1 (d) 7
13. 13.What is the smallest number you must add to 803 to make it divisible by 10? (a) 6 (b) 7 (c) 3 (d) 8
14. 14.What is the smallest number you must add to 182 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 4
15. 15.What is the smallest number you must add to 681 to make it divisible by 9? (a) 4 (b) 2 (c) 6 (d) 3
16. 16.What is the remainder when 74 is divided by 3? (a) 0 (b) 1 (c) 2
17. 17.Which of these numbers is divisible by 3? (a) 807 (b) 806 (c) 812 (d) 809
18. 18.Which of these numbers is divisible by 9? (a) 5132 (b) 5114 (c) 5128 (d) 5121
19. 19.Which of these numbers is divisible by 4? (a) 1652 (b) 1657 (c) 1649 (d) 1655
20. 20.What is the remainder when 275 is divided by 12? (a) 11 (b) 9 (c) 1 (d) 10
21. 21.Which of these numbers is divisible by 4? (a) 3254 (b) 3257 (c) 3262 (d) 3256
22. 22.Which of these numbers is divisible by 8? (a) 3604 (b) 3595 (c) 3609 (d) 3600
23. 23.What is the smallest number you must add to 367 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
24. 24.What is the units digit of 4^13? (a) 3 (b) 6 (c) 4 (d) 5
25. 25.Which of these numbers is divisible by 3? (a) 2071 (b) 2074 (c) 2070 (d) 2069
26. 26.What is the smallest number you must add to 127 to make it divisible by 3? (a) 2 (b) 1
27. 27.What is the units digit of 7^43? (a) 1 (b) 9 (c) 7 (d) 3
28. 28.What is the smallest number you must add to 154 to make it divisible by 4? (a) 2 (b) 1 (c) 3
29. 29.Which of these numbers is divisible by 3? (a) 499 (b) 494 (c) 496 (d) 495
30. 30.What is the units digit of 8^10? (a) 2 (b) 4 (c) 6 (d) 8
31. 31.Which of these numbers is divisible by 8? (a) 6330 (b) 6316 (c) 6320 (d) 6327
32. 32.Which of these numbers is divisible by 3? (a) 527 (b) 520 (c) 522 (d) 523
33. 33.Which of these numbers is divisible by 9? (a) 2951 (b) 2927 (c) 2941 (d) 2934
34. 34.What is the remainder when 119 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
35. 35.What is the smallest number you must add to 77 to make it divisible by 3? (a) 2 (b) 1
36. 36.What is the units digit of 3^10? (a) 7 (b) 9 (c) 3 (d) 1
37. 37.Which of these numbers is divisible by 3? (a) 1215 (b) 1213 (c) 1220 (d) 1217
38. 38.Which of these numbers is divisible by 3? (a) 642 (b) 646 (c) 643 (d) 640
39. 39.What is the remainder when 756 is divided by 10? (a) 7 (b) 6 (c) 5 (d) 4
40. 40.What is the remainder when 226 is divided by 9? (a) 0 (b) 2 (c) 8 (d) 1

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