Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 790 · 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 87 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 2
2. 2.What is the remainder when 138 is divided by 8? (a) 6 (b) 2 (c) 3 (d) 1
3. 3.What is the remainder when 297 is divided by 8? (a) 2 (b) 0 (c) 1 (d) 7
4. 4.What is the remainder when 225 is divided by 7? (a) 2 (b) 1 (c) 0 (d) 6
5. 5.What is the remainder when 219 is divided by 11? (a) 8 (b) 10 (c) 9 (d) 1
6. 6.Which of these numbers is divisible by 4? (a) 608 (b) 610 (c) 607 (d) 613
7. 7.Which of these numbers is divisible by 9? (a) 1366 (b) 1368 (c) 1384 (d) 1369
8. 8.What is the smallest number you must add to 777 to make it divisible by 11? (a) 3 (b) 7 (c) 4 (d) 5
9. 9.Which of these numbers is divisible by 9? (a) 2613 (b) 2621 (c) 2635 (d) 2619
10. 10.What is the units digit of 3^44? (a) 9 (b) 7 (c) 3 (d) 1
11. 11.What is the remainder when 237 is divided by 7? (a) 1 (b) 5 (c) 6 (d) 4
12. 12.What is the remainder when 611 is divided by 10? (a) 0 (b) 2 (c) 1 (d) 9
13. 13.What is the smallest number you must add to 488 to make it divisible by 9? (a) 6 (b) 7 (c) 8 (d) 2
14. 14.Which of these numbers is divisible by 11? (a) 7491 (b) 7482 (c) 7493 (d) 7512
15. 15.What is the smallest number you must add to 356 to make it divisible by 6? (a) 2 (b) 5 (c) 4 (d) 3
16. 16.Which of these numbers is divisible by 3? (a) 1500 (b) 1501 (c) 1504 (d) 1499
17. 17.Which of these numbers is divisible by 4? (a) 194 (b) 202 (c) 197 (d) 196
18. 18.What is the units digit of 9^16? (a) 2 (b) 1 (c) 0 (d) 9
19. 19.What is the units digit of 3^45? (a) 9 (b) 7 (c) 1 (d) 3
20. 20.What is the remainder when 375 is divided by 9? (a) 5 (b) 7 (c) 6 (d) 3
21. 21.What is the smallest number you must add to 306 to make it divisible by 7? (a) 2 (b) 5 (c) 3 (d) 1
22. 22.Which of these numbers is divisible by 4? (a) 447 (b) 439 (c) 440 (d) 441
23. 23.What is the units digit of 4^22? (a) 4 (b) 5 (c) 7 (d) 6
24. 24.What is the remainder when 533 is divided by 10? (a) 2 (b) 7 (c) 4 (d) 3
25. 25.What is the remainder when 192 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
26. 26.What is the remainder when 319 is divided by 7? (a) 6 (b) 3 (c) 5 (d) 4
27. 27.What is the smallest number you must add to 389 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 1
28. 28.What is the remainder when 384 is divided by 7? (a) 5 (b) 6 (c) 1 (d) 4
29. 29.What is the units digit of 7^48? (a) 7 (b) 9 (c) 3 (d) 1
30. 30.What is the remainder when 77 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
31. 31.What is the smallest number you must add to 322 to make it divisible by 11? (a) 9 (b) 7 (c) 3 (d) 8
32. 32.What is the smallest number you must add to 769 to make it divisible by 12? (a) 11 (b) 10 (c) 9 (d) 1
33. 33.What is the units digit of 2^24? (a) 2 (b) 6 (c) 4 (d) 8
34. 34.What is the remainder when 654 is divided by 12? (a) 8 (b) 7 (c) 6 (d) 5
35. 35.What is the units digit of 4^14? (a) 5 (b) 7 (c) 4 (d) 6
36. 36.What is the remainder when 169 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
37. 37.What is the remainder when 187 is divided by 10? (a) 3 (b) 7 (c) 8 (d) 6
38. 38.What is the units digit of 2^38? (a) 6 (b) 8 (c) 4 (d) 2
39. 39.What is the remainder when 608 is divided by 12? (a) 4 (b) 7 (c) 8 (d) 9
40. 40.What is the remainder when 197 is divided by 10? (a) 7 (b) 8 (c) 6 (d) 3

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