Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 420 · 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 229 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 1
2. 2.Which of these numbers is divisible by 8? (a) 1151 (b) 1156 (c) 1166 (d) 1152
3. 3.What is the units digit of 7^56? (a) 7 (b) 1 (c) 3 (d) 9
4. 4.Which of these numbers is divisible by 9? (a) 1266 (b) 1250 (c) 1251 (d) 1259
5. 5.Which of these numbers is divisible by 9? (a) 1420 (b) 1403 (c) 1404 (d) 1405
6. 6.What is the remainder when 389 is divided by 11? (a) 3 (b) 4 (c) 7 (d) 5
7. 7.What is the smallest number you must add to 155 to make it divisible by 11? (a) 9 (b) 8 (c) 10 (d) 1
8. 8.Which of these numbers is divisible by 3? (a) 2443 (b) 2437 (c) 2441 (d) 2439
9. 9.What is the smallest number you must add to 124 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 1
10. 10.What is the units digit of 7^39? (a) 3 (b) 7 (c) 1 (d) 9
11. 11.Which of these numbers is divisible by 3? (a) 2405 (b) 2402 (c) 2403 (d) 2408
12. 12.What is the remainder when 217 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
13. 13.What is the remainder when 552 is divided by 11? (a) 2 (b) 3 (c) 9 (d) 1
14. 14.What is the units digit of 4^20? (a) 5 (b) 4 (c) 7 (d) 6
15. 15.What is the smallest number you must add to 52 to make it divisible by 3? (a) 1 (b) 2
16. 16.What is the units digit of 9^22? (a) 2 (b) 9 (c) 1 (d) 0
17. 17.What is the units digit of 4^43? (a) 4 (b) 6 (c) 5 (d) 3
18. 18.What is the remainder when 271 is divided by 11? (a) 4 (b) 6 (c) 8 (d) 7
19. 19.Which of these numbers is divisible by 3? (a) 2125 (b) 2121 (c) 2122 (d) 2120
20. 20.What is the units digit of 3^32? (a) 3 (b) 9 (c) 7 (d) 1
21. 21.What is the smallest number you must add to 157 to make it divisible by 7? (a) 3 (b) 6 (c) 5 (d) 4
22. 22.Which of these numbers is divisible by 8? (a) 6759 (b) 6739 (c) 6744 (d) 6746
23. 23.What is the smallest number you must add to 942 to make it divisible by 12? (a) 6 (b) 7 (c) 8 (d) 5
24. 24.Which of these numbers is divisible by 8? (a) 4685 (b) 4669 (c) 4675 (d) 4672
25. 25.Which of these numbers is divisible by 11? (a) 599 (b) 590 (c) 579 (d) 583
26. 26.What is the remainder when 91 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
27. 27.Which of these numbers is divisible by 6? (a) 4595 (b) 4584 (c) 4582 (d) 4586
28. 28.What is the units digit of 2^13? (a) 6 (b) 2 (c) 4 (d) 8
29. 29.What is the remainder when 606 is divided by 10? (a) 7 (b) 4 (c) 6 (d) 5
30. 30.Which of these numbers is divisible by 11? (a) 8142 (b) 8163 (c) 8151 (d) 8157
31. 31.What is the units digit of 2^14? (a) 6 (b) 4 (c) 8 (d) 2
32. 32.What is the remainder when 67 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
33. 33.What is the smallest number you must add to 266 to make it divisible by 8? (a) 2 (b) 5 (c) 7 (d) 6
34. 34.What is the units digit of 7^24? (a) 7 (b) 1 (c) 9 (d) 3
35. 35.What is the smallest number you must add to 79 to make it divisible by 4? (a) 1 (b) 3 (c) 2
36. 36.Which of these numbers is divisible by 9? (a) 3905 (b) 3894 (c) 3887 (d) 3888
37. 37.What is the units digit of 8^20? (a) 8 (b) 2 (c) 6 (d) 4
38. 38.What is the smallest number you must add to 875 to make it divisible by 11? (a) 5 (b) 7 (c) 4 (d) 6
39. 39.Which of these numbers is divisible by 4? (a) 2932 (b) 2939 (c) 2935 (d) 2931
40. 40.What is the remainder when 452 is divided by 10? (a) 8 (b) 1 (c) 2 (d) 3

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