Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 729 · 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 385 to make it divisible by 8? (a) 7 (b) 5 (c) 6 (d) 1
2. 2.What is the smallest number you must add to 168 to make it divisible by 11? (a) 7 (b) 9 (c) 8 (d) 3
3. 3.What is the smallest number you must add to 162 to make it divisible by 11? (a) 8 (b) 4 (c) 2 (d) 3
4. 4.Which of these numbers is divisible by 8? (a) 2826 (b) 2824 (c) 2818 (d) 2838
5. 5.Which of these numbers is divisible by 8? (a) 5137 (b) 5151 (c) 5130 (d) 5136
6. 6.What is the smallest number you must add to 566 to make it divisible by 7? (a) 3 (b) 6 (c) 1 (d) 2
7. 7.What is the units digit of 9^21? (a) 9 (b) 7 (c) 1 (d) 8
8. 8.Which of these numbers is divisible by 3? (a) 1238 (b) 1233 (c) 1231 (d) 1234
9. 9.What is the units digit of 8^54? (a) 8 (b) 6 (c) 2 (d) 4
10. 10.Which of these numbers is divisible by 11? (a) 2087 (b) 2099 (c) 2077 (d) 2079
11. 11.What is the units digit of 2^25? (a) 4 (b) 2 (c) 6 (d) 8
12. 12.What is the units digit of 9^43? (a) 7 (b) 8 (c) 9 (d) 1
13. 13.What is the smallest number you must add to 237 to make it divisible by 12? (a) 2 (b) 9 (c) 4 (d) 3
14. 14.What is the units digit of 9^58? (a) 9 (b) 2 (c) 0 (d) 1
15. 15.What is the remainder when 442 is divided by 10? (a) 2 (b) 8 (c) 1 (d) 3
16. 16.What is the units digit of 9^57? (a) 8 (b) 7 (c) 1 (d) 9
17. 17.Which of these numbers is divisible by 3? (a) 2380 (b) 2382 (c) 2383 (d) 2387
18. 18.Which of these numbers is divisible by 9? (a) 7318 (b) 7326 (c) 7340 (d) 7328
19. 19.What is the remainder when 128 is divided by 3? (a) 1 (b) 0 (c) 2
20. 20.What is the smallest number you must add to 106 to make it divisible by 6? (a) 3 (b) 1 (c) 2 (d) 4
21. 21.What is the remainder when 715 is divided by 10? (a) 4 (b) 7 (c) 5 (d) 6
22. 22.What is the units digit of 2^40? (a) 4 (b) 2 (c) 8 (d) 6
23. 23.What is the units digit of 3^20? (a) 3 (b) 7 (c) 1 (d) 9
24. 24.What is the smallest number you must add to 73 to make it divisible by 3? (a) 2 (b) 1
25. 25.What is the smallest number you must add to 100 to make it divisible by 9? (a) 8 (b) 7 (c) 1 (d) 6
26. 26.Which of these numbers is divisible by 8? (a) 7063 (b) 7065 (c) 7053 (d) 7056
27. 27.What is the units digit of 2^30? (a) 6 (b) 2 (c) 4 (d) 8
28. 28.What is the smallest number you must add to 140 to make it divisible by 11? (a) 2 (b) 3 (c) 4 (d) 8
29. 29.Which of these numbers is divisible by 6? (a) 619 (b) 612 (c) 615 (d) 609
30. 30.Which of these numbers is divisible by 11? (a) 898 (b) 908 (c) 890 (d) 891
31. 31.What is the smallest number you must add to 107 to make it divisible by 8? (a) 5 (b) 6 (c) 4 (d) 3
32. 32.What is the units digit of 2^32? (a) 4 (b) 8 (c) 6 (d) 2
33. 33.What is the units digit of 3^52? (a) 3 (b) 1 (c) 9 (d) 7
34. 34.What is the remainder when 86 is divided by 8? (a) 5 (b) 7 (c) 6 (d) 2
35. 35.What is the remainder when 87 is divided by 5? (a) 1 (b) 3 (c) 2 (d) 4
36. 36.What is the smallest number you must add to 271 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
37. 37.What is the remainder when 322 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
38. 38.What is the remainder when 51 is divided by 5? (a) 4 (b) 0 (c) 1 (d) 2
39. 39.Which of these numbers is divisible by 8? (a) 828 (b) 814 (c) 816 (d) 819
40. 40.What is the units digit of 3^57? (a) 9 (b) 7 (c) 3 (d) 1

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