Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 196 · 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 units digit of 3^14? (a) 9 (b) 1 (c) 3 (d) 7
2. 2.What is the remainder when 334 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
3. 3.What is the remainder when 249 is divided by 10? (a) 9 (b) 1 (c) 7 (d) 8
4. 4.What is the units digit of 9^10? (a) 9 (b) 2 (c) 0 (d) 1
5. 5.What is the remainder when 147 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
6. 6.What is the units digit of 4^22? (a) 7 (b) 5 (c) 6 (d) 4
7. 7.What is the smallest number you must add to 141 to make it divisible by 8? (a) 2 (b) 4 (c) 3 (d) 5
8. 8.Which of these numbers is divisible by 6? (a) 1589 (b) 1578 (c) 1575 (d) 1581
9. 9.What is the remainder when 328 is divided by 7? (a) 4 (b) 1 (c) 6 (d) 5
10. 10.Which of these numbers is divisible by 6? (a) 2406 (b) 2408 (c) 2403 (d) 2416
11. 11.Which of these numbers is divisible by 11? (a) 7734 (b) 7731 (c) 7712 (d) 7722
12. 12.Which of these numbers is divisible by 3? (a) 2330 (b) 2332 (c) 2326 (d) 2328
13. 13.Which of these numbers is divisible by 8? (a) 4703 (b) 4694 (c) 4684 (d) 4688
14. 14.What is the units digit of 7^13? (a) 9 (b) 7 (c) 3 (d) 1
15. 15.What is the smallest number you must add to 581 to make it divisible by 9? (a) 4 (b) 5 (c) 6 (d) 3
16. 16.Which of these numbers is divisible by 11? (a) 1738 (b) 1744 (c) 1732 (d) 1757
17. 17.What is the remainder when 621 is divided by 10? (a) 1 (b) 0 (c) 9 (d) 2
18. 18.What is the remainder when 135 is divided by 6? (a) 4 (b) 2 (c) 5 (d) 3
19. 19.What is the units digit of 9^34? (a) 9 (b) 0 (c) 2 (d) 1
20. 20.What is the remainder when 72 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
21. 21.Which of these numbers is divisible by 4? (a) 1151 (b) 1153 (c) 1158 (d) 1152
22. 22.Which of these numbers is divisible by 4? (a) 945 (b) 940 (c) 939 (d) 941
23. 23.What is the smallest number you must add to 940 to make it divisible by 12? (a) 8 (b) 7 (c) 4 (d) 9
24. 24.What is the units digit of 9^52? (a) 2 (b) 9 (c) 1 (d) 0
25. 25.What is the remainder when 329 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
26. 26.What is the units digit of 8^50? (a) 8 (b) 4 (c) 6 (d) 2
27. 27.What is the remainder when 725 is divided by 9? (a) 6 (b) 7 (c) 5 (d) 4
28. 28.What is the units digit of 3^36? (a) 9 (b) 1 (c) 7 (d) 3
29. 29.What is the remainder when 131 is divided by 5? (a) 0 (b) 4 (c) 2 (d) 1
30. 30.Which of these numbers is divisible by 3? (a) 2567 (b) 2564 (c) 2560 (d) 2562
31. 31.What is the units digit of 3^17? (a) 1 (b) 9 (c) 3 (d) 7
32. 32.What is the remainder when 115 is divided by 3? (a) 2 (b) 1 (c) 0
33. 33.Which of these numbers is divisible by 11? (a) 8558 (b) 8556 (c) 8565 (d) 8571
34. 34.Which of these numbers is divisible by 11? (a) 9832 (b) 9854 (c) 9836 (d) 9834
35. 35.Which of these numbers is divisible by 4? (a) 257 (b) 256 (c) 263 (d) 253
36. 36.What is the smallest number you must add to 525 to make it divisible by 8? (a) 5 (b) 2 (c) 3 (d) 4
37. 37.Which of these numbers is divisible by 4? (a) 579 (b) 573 (c) 569 (d) 572
38. 38.Which of these numbers is divisible by 11? (a) 4029 (b) 4037 (c) 4053 (d) 4043
39. 39.What is the smallest number you must add to 104 to make it divisible by 3? (a) 1 (b) 2
40. 40.What is the smallest number you must add to 226 to make it divisible by 4? (a) 3 (b) 1 (c) 2

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