Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 10 · 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 294 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
2. 2.What is the remainder when 372 is divided by 11? (a) 9 (b) 8 (c) 2 (d) 10
3. 3.Which of these numbers is divisible by 9? (a) 5391 (b) 5393 (c) 5384 (d) 5406
4. 4.What is the remainder when 498 is divided by 9? (a) 6 (b) 2 (c) 3 (d) 4
5. 5.What is the units digit of 4^47? (a) 6 (b) 3 (c) 4 (d) 5
6. 6.What is the units digit of 8^34? (a) 4 (b) 6 (c) 8 (d) 2
7. 7.Which of these numbers is divisible by 9? (a) 5118 (b) 5122 (c) 5137 (d) 5121
8. 8.Which of these numbers is divisible by 6? (a) 1033 (b) 1022 (c) 1026 (d) 1027
9. 9.Which of these numbers is divisible by 4? (a) 3511 (b) 3518 (c) 3512 (d) 3515
10. 10.What is the smallest number you must add to 282 to make it divisible by 4? (a) 1 (b) 3 (c) 2
11. 11.Which of these numbers is divisible by 4? (a) 864 (b) 871 (c) 866 (d) 861
12. 12.What is the smallest number you must add to 506 to make it divisible by 9? (a) 6 (b) 8 (c) 7 (d) 2
13. 13.What is the remainder when 616 is divided by 10? (a) 6 (b) 5 (c) 7 (d) 4
14. 14.Which of these numbers is divisible by 6? (a) 3912 (b) 3920 (c) 3917 (d) 3910
15. 15.What is the smallest number you must add to 124 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
16. 16.What is the units digit of 8^44? (a) 8 (b) 6 (c) 2 (d) 4
17. 17.What is the units digit of 7^14? (a) 3 (b) 9 (c) 7 (d) 1
18. 18.What is the remainder when 213 is divided by 12? (a) 8 (b) 10 (c) 3 (d) 9
19. 19.Which of these numbers is divisible by 9? (a) 570 (b) 578 (c) 567 (d) 566
20. 20.What is the units digit of 8^36? (a) 6 (b) 4 (c) 2 (d) 8
21. 21.What is the remainder when 871 is divided by 11? (a) 9 (b) 3 (c) 2 (d) 1
22. 22.Which of these numbers is divisible by 8? (a) 1415 (b) 1400 (c) 1397 (d) 1402
23. 23.What is the remainder when 238 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
24. 24.What is the remainder when 368 is divided by 10? (a) 9 (b) 2 (c) 7 (d) 8
25. 25.What is the units digit of 8^53? (a) 8 (b) 4 (c) 2 (d) 6
26. 26.What is the smallest number you must add to 205 to make it divisible by 3? (a) 1 (b) 2
27. 27.What is the remainder when 841 is divided by 11? (a) 6 (b) 4 (c) 5 (d) 7
28. 28.Which of these numbers is divisible by 4? (a) 2156 (b) 2153 (c) 2159 (d) 2163
29. 29.What is the units digit of 4^33? (a) 6 (b) 3 (c) 5 (d) 4
30. 30.What is the smallest number you must add to 279 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
31. 31.What is the smallest number you must add to 241 to make it divisible by 10? (a) 9 (b) 1 (c) 7 (d) 8
32. 32.Which of these numbers is divisible by 3? (a) 1429 (b) 1432 (c) 1427 (d) 1428
33. 33.Which of these numbers is divisible by 4? (a) 1036 (b) 1043 (c) 1037 (d) 1033
34. 34.Which of these numbers is divisible by 11? (a) 3369 (b) 3360 (c) 3385 (d) 3366
35. 35.What is the units digit of 3^37? (a) 7 (b) 1 (c) 3 (d) 9
36. 36.What is the smallest number you must add to 389 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
37. 37.What is the smallest number you must add to 750 to make it divisible by 11? (a) 2 (b) 10 (c) 9 (d) 8
38. 38.What is the smallest number you must add to 76 to make it divisible by 6? (a) 3 (b) 1 (c) 2 (d) 4
39. 39.Which of these numbers is divisible by 4? (a) 2665 (b) 2664 (c) 2663 (d) 2671
40. 40.Which of these numbers is divisible by 4? (a) 811 (b) 804 (c) 803 (d) 807

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