Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 750 · 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 71 to make it divisible by 3? (a) 1 (b) 2
2. 2.What is the smallest number you must add to 279 to make it divisible by 4? (a) 3 (b) 2 (c) 1
3. 3.What is the smallest number you must add to 45 to make it divisible by 4? (a) 1 (b) 2 (c) 3
4. 4.What is the remainder when 279 is divided by 6? (a) 2 (b) 5 (c) 3 (d) 4
5. 5.What is the smallest number you must add to 129 to make it divisible by 4? (a) 2 (b) 1 (c) 3
6. 6.Which of these numbers is divisible by 8? (a) 788 (b) 776 (c) 780 (d) 769
7. 7.What is the remainder when 563 is divided by 12? (a) 11 (b) 1 (c) 9 (d) 10
8. 8.Which of these numbers is divisible by 9? (a) 1288 (b) 1272 (c) 1278 (d) 1282
9. 9.Which of these numbers is divisible by 8? (a) 925 (b) 912 (c) 916 (d) 909
10. 10.What is the units digit of 3^10? (a) 7 (b) 1 (c) 9 (d) 3
11. 11.What is the smallest number you must add to 314 to make it divisible by 11? (a) 4 (b) 7 (c) 5 (d) 6
12. 12.Which of these numbers is divisible by 6? (a) 2155 (b) 2151 (c) 2161 (d) 2154
13. 13.What is the remainder when 141 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
14. 14.What is the smallest number you must add to 89 to make it divisible by 3? (a) 2 (b) 1
15. 15.What is the smallest number you must add to 171 to make it divisible by 11? (a) 4 (b) 6 (c) 7 (d) 5
16. 16.What is the smallest number you must add to 279 to make it divisible by 11? (a) 4 (b) 8 (c) 6 (d) 7
17. 17.Which of these numbers is divisible by 8? (a) 2423 (b) 2426 (c) 2409 (d) 2416
18. 18.What is the units digit of 4^12? (a) 4 (b) 7 (c) 5 (d) 6
19. 19.Which of these numbers is divisible by 3? (a) 313 (b) 320 (c) 317 (d) 315
20. 20.What is the remainder when 280 is divided by 9? (a) 2 (b) 0 (c) 1 (d) 8
21. 21.What is the remainder when 711 is divided by 10? (a) 2 (b) 9 (c) 1 (d) 0
22. 22.What is the remainder when 618 is divided by 9? (a) 6 (b) 3 (c) 7 (d) 5
23. 23.What is the units digit of 3^14? (a) 9 (b) 3 (c) 1 (d) 7
24. 24.What is the smallest number you must add to 570 to make it divisible by 11? (a) 3 (b) 9 (c) 1 (d) 2
25. 25.Which of these numbers is divisible by 6? (a) 2213 (b) 2207 (c) 2202 (d) 2197
26. 26.What is the remainder when 127 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
27. 27.What is the units digit of 8^19? (a) 8 (b) 2 (c) 6 (d) 4
28. 28.What is the units digit of 4^19? (a) 5 (b) 3 (c) 4 (d) 6
29. 29.What is the smallest number you must add to 106 to make it divisible by 3? (a) 1 (b) 2
30. 30.What is the smallest number you must add to 206 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
31. 31.What is the units digit of 3^22? (a) 7 (b) 1 (c) 3 (d) 9
32. 32.Which of these numbers is divisible by 8? (a) 1102 (b) 1109 (c) 1094 (d) 1096
33. 33.What is the smallest number you must add to 112 to make it divisible by 6? (a) 3 (b) 2 (c) 1 (d) 4
34. 34.What is the units digit of 9^21? (a) 1 (b) 8 (c) 9 (d) 7
35. 35.What is the smallest number you must add to 47 to make it divisible by 3? (a) 2 (b) 1
36. 36.What is the smallest number you must add to 273 to make it divisible by 4? (a) 2 (b) 1 (c) 3
37. 37.What is the smallest number you must add to 318 to make it divisible by 4? (a) 1 (b) 3 (c) 2
38. 38.What is the units digit of 2^22? (a) 6 (b) 2 (c) 8 (d) 4
39. 39.What is the remainder when 208 is divided by 7? (a) 4 (b) 6 (c) 2 (d) 5
40. 40.What is the units digit of 8^20? (a) 8 (b) 2 (c) 6 (d) 4

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