Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 646 · 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.Which of these numbers is divisible by 11? (a) 3127 (b) 3120 (c) 3141 (d) 3124
2. 2.Which of these numbers is divisible by 4? (a) 3515 (b) 3517 (c) 3512 (d) 3510
3. 3.Which of these numbers is divisible by 4? (a) 1515 (b) 1517 (c) 1522 (d) 1516
4. 4.What is the units digit of 3^13? (a) 1 (b) 9 (c) 7 (d) 3
5. 5.Which of these numbers is divisible by 4? (a) 486 (b) 483 (c) 480 (d) 478
6. 6.What is the units digit of 4^24? (a) 4 (b) 6 (c) 5 (d) 7
7. 7.What is the units digit of 9^19? (a) 8 (b) 1 (c) 7 (d) 9
8. 8.What is the units digit of 7^15? (a) 3 (b) 1 (c) 7 (d) 9
9. 9.What is the smallest number you must add to 84 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
10. 10.What is the smallest number you must add to 887 to make it divisible by 11? (a) 5 (b) 3 (c) 7 (d) 4
11. 11.What is the units digit of 2^54? (a) 2 (b) 8 (c) 4 (d) 6
12. 12.Which of these numbers is divisible by 4? (a) 1333 (b) 1337 (c) 1330 (d) 1332
13. 13.Which of these numbers is divisible by 4? (a) 3274 (b) 3277 (c) 3269 (d) 3272
14. 14.Which of these numbers is divisible by 11? (a) 1826 (b) 1831 (c) 1847 (d) 1821
15. 15.What is the units digit of 3^16? (a) 3 (b) 1 (c) 9 (d) 7
16. 16.Which of these numbers is divisible by 6? (a) 1662 (b) 1660 (c) 1663 (d) 1669
17. 17.What is the smallest number you must add to 495 to make it divisible by 12? (a) 3 (b) 10 (c) 9 (d) 8
18. 18.What is the smallest number you must add to 432 to make it divisible by 7? (a) 5 (b) 3 (c) 2 (d) 1
19. 19.Which of these numbers is divisible by 9? (a) 5670 (b) 5668 (c) 5672 (d) 5685
20. 20.What is the remainder when 212 is divided by 3? (a) 0 (b) 2 (c) 1
21. 21.What is the remainder when 109 is divided by 9? (a) 2 (b) 0 (c) 1 (d) 8
22. 22.What is the smallest number you must add to 281 to make it divisible by 4? (a) 1 (b) 2 (c) 3
23. 23.What is the units digit of 3^18? (a) 7 (b) 3 (c) 1 (d) 9
24. 24.What is the units digit of 3^22? (a) 3 (b) 7 (c) 1 (d) 9
25. 25.What is the smallest number you must add to 169 to make it divisible by 3? (a) 2 (b) 1
26. 26.What is the units digit of 4^10? (a) 5 (b) 4 (c) 6 (d) 7
27. 27.What is the smallest number you must add to 966 to make it divisible by 12? (a) 7 (b) 5 (c) 6 (d) 8
28. 28.Which of these numbers is divisible by 8? (a) 3858 (b) 3853 (c) 3865 (d) 3856
29. 29.What is the remainder when 211 is divided by 7? (a) 2 (b) 0 (c) 1 (d) 6
30. 30.Which of these numbers is divisible by 8? (a) 4682 (b) 4688 (c) 4699 (d) 4690
31. 31.Which of these numbers is divisible by 8? (a) 7121 (b) 7128 (c) 7139 (d) 7134
32. 32.Which of these numbers is divisible by 9? (a) 978 (b) 971 (c) 972 (d) 987
33. 33.What is the remainder when 583 is divided by 8? (a) 1 (b) 7 (c) 6 (d) 5
34. 34.What is the smallest number you must add to 267 to make it divisible by 10? (a) 4 (b) 2 (c) 3 (d) 7
35. 35.What is the units digit of 4^36? (a) 6 (b) 4 (c) 7 (d) 5
36. 36.Which of these numbers is divisible by 3? (a) 718 (b) 720 (c) 725 (d) 721
37. 37.What is the smallest number you must add to 195 to make it divisible by 10? (a) 4 (b) 6 (c) 7 (d) 5
38. 38.What is the remainder when 573 is divided by 8? (a) 5 (b) 6 (c) 4 (d) 3
39. 39.What is the units digit of 9^10? (a) 0 (b) 9 (c) 2 (d) 1
40. 40.Which of these numbers is divisible by 6? (a) 256 (b) 246 (c) 248 (d) 243

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