Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 48 · 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 4? (a) 1067 (b) 1060 (c) 1058 (d) 1063
2. 2.What is the smallest number you must add to 178 to make it divisible by 3? (a) 2 (b) 1
3. 3.What is the smallest number you must add to 617 to make it divisible by 8? (a) 5 (b) 6 (c) 1 (d) 7
4. 4.Which of these numbers is divisible by 6? (a) 3023 (b) 3032 (c) 3024 (d) 3026
5. 5.What is the units digit of 8^52? (a) 8 (b) 6 (c) 4 (d) 2
6. 6.Which of these numbers is divisible by 4? (a) 261 (b) 257 (c) 254 (d) 256
7. 7.What is the remainder when 157 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
8. 8.What is the smallest number you must add to 664 to make it divisible by 12? (a) 4 (b) 7 (c) 9 (d) 8
9. 9.What is the smallest number you must add to 536 to make it divisible by 10? (a) 5 (b) 3 (c) 6 (d) 4
10. 10.Which of these numbers is divisible by 8? (a) 980 (b) 968 (c) 974 (d) 962
11. 11.What is the units digit of 2^56? (a) 2 (b) 6 (c) 4 (d) 8
12. 12.What is the smallest number you must add to 116 to make it divisible by 3? (a) 2 (b) 1
13. 13.What is the units digit of 4^53? (a) 6 (b) 3 (c) 5 (d) 4
14. 14.What is the remainder when 135 is divided by 11? (a) 4 (b) 8 (c) 2 (d) 3
15. 15.Which of these numbers is divisible by 3? (a) 2524 (b) 2522 (c) 2523 (d) 2527
16. 16.What is the remainder when 281 is divided by 7? (a) 1 (b) 0 (c) 2 (d) 6
17. 17.What is the units digit of 8^47? (a) 4 (b) 2 (c) 6 (d) 8
18. 18.Which of these numbers is divisible by 11? (a) 6440 (b) 6419 (c) 6424 (d) 6425
19. 19.Which of these numbers is divisible by 8? (a) 4672 (b) 4687 (c) 4669 (d) 4674
20. 20.What is the units digit of 4^19? (a) 3 (b) 4 (c) 6 (d) 5
21. 21.Which of these numbers is divisible by 3? (a) 1056 (b) 1061 (c) 1057 (d) 1054
22. 22.What is the smallest number you must add to 368 to make it divisible by 7? (a) 4 (b) 2 (c) 3 (d) 5
23. 23.What is the units digit of 2^26? (a) 8 (b) 6 (c) 2 (d) 4
24. 24.Which of these numbers is divisible by 3? (a) 833 (b) 831 (c) 829 (d) 835
25. 25.What is the units digit of 3^25? (a) 7 (b) 3 (c) 1 (d) 9
26. 26.What is the smallest number you must add to 367 to make it divisible by 8? (a) 3 (b) 7 (c) 1 (d) 2
27. 27.What is the smallest number you must add to 419 to make it divisible by 12? (a) 11 (b) 2 (c) 3 (d) 1
28. 28.Which of these numbers is divisible by 3? (a) 232 (b) 228 (c) 230 (d) 227
29. 29.What is the remainder when 91 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
30. 30.What is the units digit of 2^39? (a) 2 (b) 4 (c) 8 (d) 6
31. 31.What is the remainder when 122 is divided by 3? (a) 0 (b) 1 (c) 2
32. 32.What is the remainder when 727 is divided by 11? (a) 0 (b) 2 (c) 10 (d) 1
33. 33.What is the units digit of 7^59? (a) 1 (b) 3 (c) 7 (d) 9
34. 34.What is the remainder when 65 is divided by 3? (a) 0 (b) 1 (c) 2
35. 35.Which of these numbers is divisible by 9? (a) 7170 (b) 7153 (c) 7163 (d) 7155
36. 36.What is the remainder when 184 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
37. 37.What is the remainder when 851 is divided by 11? (a) 4 (b) 3 (c) 5 (d) 7
38. 38.What is the remainder when 638 is divided by 12? (a) 3 (b) 10 (c) 1 (d) 2
39. 39.What is the remainder when 499 is divided by 12? (a) 8 (b) 5 (c) 7 (d) 6
40. 40.What is the smallest number you must add to 294 to make it divisible by 4? (a) 1 (b) 2 (c) 3

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