Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 927 · 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 113 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
2. 2.What is the units digit of 7^42? (a) 1 (b) 3 (c) 7 (d) 9
3. 3.What is the remainder when 523 is divided by 7? (a) 5 (b) 2 (c) 6 (d) 4
4. 4.What is the units digit of 3^23? (a) 1 (b) 7 (c) 3 (d) 9
5. 5.Which of these numbers is divisible by 4? (a) 402 (b) 407 (c) 404 (d) 411
6. 6.What is the smallest number you must add to 71 to make it divisible by 3? (a) 1 (b) 2
7. 7.What is the units digit of 8^57? (a) 8 (b) 4 (c) 2 (d) 6
8. 8.What is the smallest number you must add to 148 to make it divisible by 11? (a) 7 (b) 5 (c) 6 (d) 8
9. 9.Which of these numbers is divisible by 4? (a) 2638 (b) 2647 (c) 2643 (d) 2640
10. 10.What is the units digit of 2^25? (a) 6 (b) 4 (c) 8 (d) 2
11. 11.What is the units digit of 4^31? (a) 5 (b) 3 (c) 6 (d) 4
12. 12.Which of these numbers is divisible by 6? (a) 5357 (b) 5352 (c) 5362 (d) 5347
13. 13.What is the smallest number you must add to 185 to make it divisible by 7? (a) 5 (b) 3 (c) 6 (d) 4
14. 14.Which of these numbers is divisible by 4? (a) 1855 (b) 1856 (c) 1858 (d) 1861
15. 15.Which of these numbers is divisible by 11? (a) 9241 (b) 9233 (c) 9240 (d) 9252
16. 16.What is the smallest number you must add to 680 to make it divisible by 11? (a) 3 (b) 9 (c) 1 (d) 2
17. 17.Which of these numbers is divisible by 6? (a) 2669 (b) 2658 (c) 2661 (d) 2656
18. 18.What is the units digit of 3^56? (a) 3 (b) 7 (c) 9 (d) 1
19. 19.What is the smallest number you must add to 129 to make it divisible by 4? (a) 3 (b) 2 (c) 1
20. 20.What is the units digit of 3^31? (a) 9 (b) 1 (c) 3 (d) 7
21. 21.What is the remainder when 321 is divided by 5? (a) 0 (b) 1 (c) 2 (d) 4
22. 22.What is the units digit of 4^52? (a) 6 (b) 7 (c) 5 (d) 4
23. 23.What is the units digit of 3^10? (a) 3 (b) 7 (c) 9 (d) 1
24. 24.Which of these numbers is divisible by 3? (a) 212 (b) 213 (c) 214 (d) 217
25. 25.What is the remainder when 318 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
26. 26.What is the smallest number you must add to 536 to make it divisible by 11? (a) 4 (b) 2 (c) 3 (d) 8
27. 27.What is the remainder when 566 is divided by 10? (a) 7 (b) 4 (c) 6 (d) 5
28. 28.What is the remainder when 580 is divided by 8? (a) 6 (b) 4 (c) 5 (d) 3
29. 29.What is the remainder when 839 is divided by 11? (a) 4 (b) 2 (c) 8 (d) 3
30. 30.What is the smallest number you must add to 115 to make it divisible by 4? (a) 1 (b) 3 (c) 2
31. 31.What is the remainder when 563 is divided by 12? (a) 1 (b) 10 (c) 9 (d) 11
32. 32.Which of these numbers is divisible by 9? (a) 852 (b) 837 (c) 841 (d) 833
33. 33.What is the remainder when 201 is divided by 5? (a) 0 (b) 4 (c) 1 (d) 2
34. 34.What is the smallest number you must add to 321 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
35. 35.What is the units digit of 3^36? (a) 9 (b) 1 (c) 7 (d) 3
36. 36.Which of these numbers is divisible by 6? (a) 1201 (b) 1193 (c) 1194 (d) 1196
37. 37.What is the remainder when 500 is divided by 9? (a) 4 (b) 6 (c) 5 (d) 7
38. 38.What is the remainder when 345 is divided by 10? (a) 4 (b) 5 (c) 6 (d) 7
39. 39.What is the remainder when 696 is divided by 9? (a) 6 (b) 2 (c) 3 (d) 4
40. 40.What is the smallest number you must add to 171 to make it divisible by 7? (a) 6 (b) 3 (c) 4 (d) 5

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