Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 707 · 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 3? (a) 1717 (b) 1714 (c) 1720 (d) 1716
2. 2.What is the units digit of 4^45? (a) 6 (b) 5 (c) 3 (d) 4
3. 3.Which of these numbers is divisible by 3? (a) 685 (b) 680 (c) 682 (d) 681
4. 4.What is the smallest number you must add to 406 to make it divisible by 6? (a) 1 (b) 4 (c) 3 (d) 2
5. 5.What is the units digit of 4^37? (a) 6 (b) 4 (c) 3 (d) 5
6. 6.Which of these numbers is divisible by 11? (a) 1481 (b) 1467 (c) 1457 (d) 1463
7. 7.Which of these numbers is divisible by 9? (a) 6921 (b) 6922 (c) 6935 (d) 6917
8. 8.What is the units digit of 2^15? (a) 4 (b) 2 (c) 6 (d) 8
9. 9.What is the smallest number you must add to 725 to make it divisible by 10? (a) 6 (b) 7 (c) 5 (d) 4
10. 10.Which of these numbers is divisible by 4? (a) 700 (b) 699 (c) 705 (d) 701
11. 11.What is the remainder when 135 is divided by 11? (a) 3 (b) 2 (c) 8 (d) 4
12. 12.Which of these numbers is divisible by 11? (a) 7819 (b) 7821 (c) 7826 (d) 7838
13. 13.Which of these numbers is divisible by 11? (a) 9173 (b) 9176 (c) 9174 (d) 9189
14. 14.Which of these numbers is divisible by 3? (a) 1702 (b) 1696 (c) 1700 (d) 1698
15. 15.What is the remainder when 226 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
16. 16.What is the units digit of 8^44? (a) 8 (b) 4 (c) 6 (d) 2
17. 17.What is the smallest number you must add to 171 to make it divisible by 8? (a) 3 (b) 4 (c) 5 (d) 6
18. 18.What is the remainder when 169 is divided by 3? (a) 0 (b) 1 (c) 2
19. 19.What is the units digit of 2^37? (a) 8 (b) 6 (c) 4 (d) 2
20. 20.What is the remainder when 178 is divided by 6? (a) 3 (b) 4 (c) 2 (d) 5
21. 21.What is the units digit of 3^35? (a) 9 (b) 7 (c) 1 (d) 3
22. 22.What is the smallest number you must add to 237 to make it divisible by 7? (a) 1 (b) 3 (c) 2 (d) 6
23. 23.What is the units digit of 3^22? (a) 9 (b) 1 (c) 7 (d) 3
24. 24.What is the units digit of 2^33? (a) 4 (b) 6 (c) 8 (d) 2
25. 25.Which of these numbers is divisible by 9? (a) 6420 (b) 6426 (c) 6439 (d) 6427
26. 26.What is the smallest number you must add to 288 to make it divisible by 7? (a) 5 (b) 4 (c) 1 (d) 6
27. 27.What is the smallest number you must add to 351 to make it divisible by 11? (a) 1 (b) 10 (c) 3 (d) 2
28. 28.What is the remainder when 164 is divided by 6? (a) 3 (b) 2 (c) 4 (d) 1
29. 29.What is the smallest number you must add to 351 to make it divisible by 12? (a) 3 (b) 8 (c) 9 (d) 10
30. 30.What is the units digit of 3^58? (a) 3 (b) 9 (c) 1 (d) 7
31. 31.What is the remainder when 359 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
32. 32.What is the units digit of 8^14? (a) 4 (b) 6 (c) 8 (d) 2
33. 33.Which of these numbers is divisible by 11? (a) 4453 (b) 4470 (c) 4455 (d) 4457
34. 34.What is the remainder when 124 is divided by 3? (a) 0 (b) 1 (c) 2
35. 35.What is the remainder when 371 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 0
36. 36.What is the smallest number you must add to 637 to make it divisible by 12? (a) 9 (b) 11 (c) 1 (d) 10
37. 37.What is the smallest number you must add to 806 to make it divisible by 11? (a) 3 (b) 9 (c) 7 (d) 8
38. 38.Which of these numbers is divisible by 8? (a) 2192 (b) 2189 (c) 2203 (d) 2196
39. 39.Which of these numbers is divisible by 4? (a) 2830 (b) 2838 (c) 2834 (d) 2832
40. 40.What is the remainder when 74 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 2

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