Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 316 · 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 8? (a) 3572 (b) 3577 (c) 3561 (d) 3568
2. 2.What is the remainder when 136 is divided by 7? (a) 3 (b) 4 (c) 5 (d) 2
3. 3.Which of these numbers is divisible by 4? (a) 491 (b) 492 (c) 497 (d) 495
4. 4.Which of these numbers is divisible by 4? (a) 1191 (b) 1181 (c) 1184 (d) 1187
5. 5.What is the smallest number you must add to 178 to make it divisible by 9? (a) 2 (b) 7 (c) 1 (d) 3
6. 6.What is the smallest number you must add to 94 to make it divisible by 7? (a) 5 (b) 3 (c) 4 (d) 6
7. 7.Which of these numbers is divisible by 4? (a) 3171 (b) 3164 (c) 3167 (d) 3161
8. 8.Which of these numbers is divisible by 3? (a) 1754 (b) 1749 (c) 1748 (d) 1751
9. 9.What is the remainder when 213 is divided by 7? (a) 2 (b) 3 (c) 5 (d) 4
10. 10.Which of these numbers is divisible by 3? (a) 833 (b) 834 (c) 836 (d) 838
11. 11.What is the remainder when 189 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
12. 12.What is the smallest number you must add to 210 to make it divisible by 11? (a) 10 (b) 9 (c) 8 (d) 1
13. 13.Which of these numbers is divisible by 11? (a) 1748 (b) 1757 (c) 1749 (d) 1770
14. 14.What is the units digit of 9^14? (a) 2 (b) 9 (c) 1 (d) 0
15. 15.Which of these numbers is divisible by 6? (a) 819 (b) 809 (c) 810 (d) 811
16. 16.Which of these numbers is divisible by 8? (a) 1519 (b) 1512 (c) 1522 (d) 1511
17. 17.Which of these numbers is divisible by 4? (a) 309 (b) 314 (c) 308 (d) 307
18. 18.What is the remainder when 193 is divided by 9? (a) 5 (b) 3 (c) 4 (d) 6
19. 19.What is the units digit of 3^59? (a) 3 (b) 1 (c) 7 (d) 9
20. 20.What is the remainder when 253 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
21. 21.What is the remainder when 240 is divided by 7? (a) 1 (b) 5 (c) 3 (d) 2
22. 22.What is the remainder when 556 is divided by 11? (a) 7 (b) 8 (c) 6 (d) 5
23. 23.What is the remainder when 386 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 0
24. 24.Which of these numbers is divisible by 4? (a) 3387 (b) 3380 (c) 3378 (d) 3383
25. 25.What is the units digit of 7^50? (a) 3 (b) 1 (c) 9 (d) 7
26. 26.What is the units digit of 3^14? (a) 3 (b) 1 (c) 9 (d) 7
27. 27.What is the remainder when 163 is divided by 11? (a) 2 (b) 10 (c) 8 (d) 9
28. 28.What is the smallest number you must add to 46 to make it divisible by 4? (a) 2 (b) 1 (c) 3
29. 29.Which of these numbers is divisible by 8? (a) 716 (b) 706 (c) 698 (d) 704
30. 30.What is the remainder when 346 is divided by 6? (a) 4 (b) 2 (c) 3 (d) 5
31. 31.What is the remainder when 46 is divided by 3? (a) 0 (b) 1 (c) 2
32. 32.Which of these numbers is divisible by 11? (a) 7311 (b) 7328 (c) 7321 (d) 7315
33. 33.What is the remainder when 251 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
34. 34.What is the smallest number you must add to 116 to make it divisible by 3? (a) 1 (b) 2
35. 35.Which of these numbers is divisible by 11? (a) 8872 (b) 8877 (c) 8882 (d) 8892
36. 36.What is the smallest number you must add to 85 to make it divisible by 4? (a) 1 (b) 3 (c) 2
37. 37.What is the smallest number you must add to 100 to make it divisible by 3? (a) 1 (b) 2
38. 38.What is the smallest number you must add to 139 to make it divisible by 3? (a) 1 (b) 2
39. 39.What is the smallest number you must add to 589 to make it divisible by 9? (a) 7 (b) 6 (c) 4 (d) 5
40. 40.Which of these numbers is divisible by 4? (a) 3067 (b) 3068 (c) 3070 (d) 3075

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