Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 68 · 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 292 to make it divisible by 12? (a) 8 (b) 9 (c) 4 (d) 7
2. 2.What is the remainder when 410 is divided by 7? (a) 3 (b) 4 (c) 6 (d) 5
3. 3.What is the units digit of 8^22? (a) 4 (b) 6 (c) 2 (d) 8
4. 4.Which of these numbers is divisible by 9? (a) 2817 (b) 2820 (c) 2812 (d) 2831
5. 5.What is the remainder when 241 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
6. 6.What is the remainder when 86 is divided by 6? (a) 4 (b) 3 (c) 2 (d) 1
7. 7.What is the units digit of 3^52? (a) 3 (b) 1 (c) 9 (d) 7
8. 8.What is the units digit of 3^15? (a) 1 (b) 3 (c) 7 (d) 9
9. 9.What is the remainder when 279 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
10. 10.Which of these numbers is divisible by 6? (a) 1622 (b) 1629 (c) 1620 (d) 1616
11. 11.What is the units digit of 3^51? (a) 1 (b) 9 (c) 3 (d) 7
12. 12.What is the units digit of 3^43? (a) 9 (b) 1 (c) 3 (d) 7
13. 13.Which of these numbers is divisible by 4? (a) 3058 (b) 3054 (c) 3056 (d) 3061
14. 14.What is the smallest number you must add to 199 to make it divisible by 4? (a) 3 (b) 2 (c) 1
15. 15.Which of these numbers is divisible by 3? (a) 582 (b) 583 (c) 580 (d) 587
16. 16.Which of these numbers is divisible by 8? (a) 2863 (b) 2847 (c) 2848 (d) 2849
17. 17.What is the remainder when 298 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
18. 18.Which of these numbers is divisible by 6? (a) 886 (b) 888 (c) 890 (d) 897
19. 19.What is the remainder when 813 is divided by 11? (a) 1 (b) 8 (c) 9 (d) 10
20. 20.Which of these numbers is divisible by 6? (a) 3954 (b) 3958 (c) 3953 (d) 3961
21. 21.What is the remainder when 411 is divided by 10? (a) 9 (b) 2 (c) 1 (d) 0
22. 22.Which of these numbers is divisible by 4? (a) 3015 (b) 3012 (c) 3017 (d) 3010
23. 23.What is the remainder when 685 is divided by 11? (a) 2 (b) 8 (c) 4 (d) 3
24. 24.What is the remainder when 823 is divided by 11? (a) 10 (b) 9 (c) 8 (d) 2
25. 25.What is the units digit of 8^24? (a) 6 (b) 8 (c) 4 (d) 2
26. 26.What is the smallest number you must add to 43 to make it divisible by 3? (a) 2 (b) 1
27. 27.What is the smallest number you must add to 436 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 1
28. 28.Which of these numbers is divisible by 9? (a) 2864 (b) 2885 (c) 2871 (d) 2879
29. 29.What is the smallest number you must add to 74 to make it divisible by 6? (a) 3 (b) 5 (c) 4 (d) 2
30. 30.What is the remainder when 218 is divided by 3? (a) 0 (b) 1 (c) 2
31. 31.What is the remainder when 433 is divided by 10? (a) 7 (b) 2 (c) 3 (d) 4
32. 32.What is the smallest number you must add to 129 to make it divisible by 9? (a) 6 (b) 3 (c) 5 (d) 7
33. 33.What is the smallest number you must add to 438 to make it divisible by 12? (a) 6 (b) 5 (c) 7 (d) 8
34. 34.What is the smallest number you must add to 297 to make it divisible by 6? (a) 4 (b) 5 (c) 2 (d) 3
35. 35.What is the units digit of 3^13? (a) 1 (b) 3 (c) 9 (d) 7
36. 36.What is the remainder when 182 is divided by 6? (a) 2 (b) 1 (c) 4 (d) 3
37. 37.Which of these numbers is divisible by 4? (a) 3207 (b) 3197 (c) 3201 (d) 3200
38. 38.What is the units digit of 3^33? (a) 9 (b) 3 (c) 1 (d) 7
39. 39.Which of these numbers is divisible by 4? (a) 3294 (b) 3290 (c) 3285 (d) 3288
40. 40.What is the remainder when 272 is divided by 11? (a) 9 (b) 3 (c) 8 (d) 7

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