Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 419 · 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 remainder when 605 is divided by 12? (a) 4 (b) 7 (c) 5 (d) 6
2. 2.What is the smallest number you must add to 384 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
3. 3.Which of these numbers is divisible by 4? (a) 1409 (b) 1408 (c) 1406 (d) 1415
4. 4.Which of these numbers is divisible by 3? (a) 2558 (b) 2560 (c) 2556 (d) 2554
5. 5.What is the smallest number you must add to 347 to make it divisible by 10? (a) 4 (b) 3 (c) 7 (d) 2
6. 6.What is the units digit of 3^46? (a) 1 (b) 3 (c) 7 (d) 9
7. 7.Which of these numbers is divisible by 11? (a) 4614 (b) 4620 (c) 4629 (d) 4634
8. 8.Which of these numbers is divisible by 4? (a) 2370 (b) 2368 (c) 2375 (d) 2365
9. 9.What is the remainder when 68 is divided by 3? (a) 0 (b) 2 (c) 1
10. 10.What is the remainder when 337 is divided by 7? (a) 0 (b) 1 (c) 2 (d) 6
11. 11.What is the units digit of 3^50? (a) 3 (b) 7 (c) 1 (d) 9
12. 12.What is the smallest number you must add to 517 to make it divisible by 10? (a) 7 (b) 3 (c) 4 (d) 2
13. 13.What is the remainder when 49 is divided by 3? (a) 1 (b) 2 (c) 0
14. 14.What is the smallest number you must add to 89 to make it divisible by 3? (a) 1 (b) 2
15. 15.What is the units digit of 9^44? (a) 1 (b) 0 (c) 9 (d) 2
16. 16.What is the remainder when 106 is divided by 3? (a) 0 (b) 2 (c) 1
17. 17.What is the smallest number you must add to 97 to make it divisible by 4? (a) 3 (b) 2 (c) 1
18. 18.Which of these numbers is divisible by 3? (a) 416 (b) 411 (c) 413 (d) 410
19. 19.What is the smallest number you must add to 134 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
20. 20.What is the units digit of 3^43? (a) 3 (b) 7 (c) 1 (d) 9
21. 21.Which of these numbers is divisible by 6? (a) 681 (b) 691 (c) 686 (d) 684
22. 22.What is the units digit of 8^43? (a) 2 (b) 6 (c) 4 (d) 8
23. 23.What is the remainder when 322 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
24. 24.What is the units digit of 7^37? (a) 1 (b) 7 (c) 9 (d) 3
25. 25.What is the smallest number you must add to 217 to make it divisible by 6? (a) 1 (b) 5 (c) 4 (d) 3
26. 26.What is the units digit of 3^32? (a) 1 (b) 7 (c) 3 (d) 9
27. 27.What is the units digit of 3^23? (a) 1 (b) 3 (c) 9 (d) 7
28. 28.Which of these numbers is divisible by 3? (a) 2521 (b) 2519 (c) 2525 (d) 2520
29. 29.Which of these numbers is divisible by 4? (a) 2207 (b) 2199 (c) 2200 (d) 2201
30. 30.What is the remainder when 239 is divided by 3? (a) 1 (b) 0 (c) 2
31. 31.What is the units digit of 3^37? (a) 1 (b) 9 (c) 3 (d) 7
32. 32.What is the units digit of 7^48? (a) 7 (b) 3 (c) 1 (d) 9
33. 33.What is the remainder when 161 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 5
34. 34.Which of these numbers is divisible by 6? (a) 4689 (b) 4693 (c) 4686 (d) 4683
35. 35.What is the units digit of 9^34? (a) 1 (b) 0 (c) 2 (d) 9
36. 36.Which of these numbers is divisible by 8? (a) 6928 (b) 6938 (c) 6927 (d) 6935
37. 37.Which of these numbers is divisible by 11? (a) 7774 (b) 7778 (c) 7764 (d) 7766
38. 38.Which of these numbers is divisible by 4? (a) 3396 (b) 3397 (c) 3403 (d) 3395
39. 39.Which of these numbers is divisible by 9? (a) 3158 (b) 3175 (c) 3166 (d) 3159
40. 40.What is the smallest number you must add to 61 to make it divisible by 4? (a) 3 (b) 1 (c) 2

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