Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 908 · 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 322 to make it divisible by 9? (a) 3 (b) 1 (c) 7 (d) 2
2. 2.What is the remainder when 148 is divided by 3? (a) 1 (b) 0 (c) 2
3. 3.What is the remainder when 139 is divided by 3? (a) 0 (b) 2 (c) 1
4. 4.What is the smallest number you must add to 114 to make it divisible by 7? (a) 6 (b) 4 (c) 2 (d) 5
5. 5.What is the smallest number you must add to 211 to make it divisible by 8? (a) 3 (b) 4 (c) 5 (d) 6
6. 6.What is the units digit of 8^31? (a) 8 (b) 4 (c) 6 (d) 2
7. 7.What is the smallest number you must add to 210 to make it divisible by 4? (a) 1 (b) 3 (c) 2
8. 8.What is the smallest number you must add to 236 to make it divisible by 12? (a) 5 (b) 8 (c) 3 (d) 4
9. 9.What is the smallest number you must add to 447 to make it divisible by 10? (a) 3 (b) 7 (c) 2 (d) 4
10. 10.Which of these numbers is divisible by 4? (a) 700 (b) 701 (c) 699 (d) 705
11. 11.Which of these numbers is divisible by 11? (a) 8439 (b) 8452 (c) 8448 (d) 8465
12. 12.What is the smallest number you must add to 433 to make it divisible by 6? (a) 1 (b) 3 (c) 5 (d) 4
13. 13.Which of these numbers is divisible by 4? (a) 3188 (b) 3185 (c) 3193 (d) 3191
14. 14.What is the units digit of 7^15? (a) 1 (b) 9 (c) 7 (d) 3
15. 15.What is the smallest number you must add to 547 to make it divisible by 7? (a) 1 (b) 5 (c) 4 (d) 6
16. 16.What is the smallest number you must add to 366 to make it divisible by 11? (a) 7 (b) 9 (c) 3 (d) 8
17. 17.What is the remainder when 153 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
18. 18.Which of these numbers is divisible by 3? (a) 272 (b) 275 (c) 277 (d) 273
19. 19.What is the units digit of 9^25? (a) 7 (b) 8 (c) 9 (d) 1
20. 20.What is the units digit of 8^10? (a) 6 (b) 2 (c) 4 (d) 8
21. 21.What is the smallest number you must add to 422 to make it divisible by 7? (a) 2 (b) 5 (c) 6 (d) 4
22. 22.Which of these numbers is divisible by 9? (a) 2775 (b) 2787 (c) 2781 (d) 2792
23. 23.What is the smallest number you must add to 479 to make it divisible by 11? (a) 7 (b) 6 (c) 5 (d) 4
24. 24.What is the smallest number you must add to 522 to make it divisible by 7? (a) 4 (b) 3 (c) 2 (d) 5
25. 25.Which of these numbers is divisible by 11? (a) 2849 (b) 2847 (c) 2862 (d) 2851
26. 26.What is the remainder when 234 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
27. 27.What is the units digit of 9^58? (a) 1 (b) 0 (c) 2 (d) 9
28. 28.What is the remainder when 211 is divided by 5? (a) 1 (b) 2 (c) 0 (d) 4
29. 29.What is the units digit of 7^30? (a) 9 (b) 3 (c) 7 (d) 1
30. 30.Which of these numbers is divisible by 6? (a) 3191 (b) 3176 (c) 3180 (d) 3182
31. 31.What is the remainder when 254 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
32. 32.What is the remainder when 178 is divided by 12? (a) 2 (b) 9 (c) 11 (d) 10
33. 33.Which of these numbers is divisible by 8? (a) 6338 (b) 6349 (c) 6332 (d) 6336
34. 34.Which of these numbers is divisible by 4? (a) 2487 (b) 2484 (c) 2482 (d) 2489
35. 35.What is the units digit of 4^26? (a) 5 (b) 7 (c) 6 (d) 4
36. 36.Which of these numbers is divisible by 8? (a) 5215 (b) 5200 (c) 5193 (d) 5201
37. 37.What is the smallest number you must add to 137 to make it divisible by 4? (a) 2 (b) 1 (c) 3
38. 38.Which of these numbers is divisible by 4? (a) 1683 (b) 1684 (c) 1690 (d) 1687
39. 39.What is the units digit of 4^15? (a) 5 (b) 3 (c) 4 (d) 6
40. 40.What is the units digit of 7^33? (a) 1 (b) 7 (c) 3 (d) 9

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