Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 943 · 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 857 is divided by 11? (a) 9 (b) 8 (c) 1 (d) 10
2. 2.Which of these numbers is divisible by 6? (a) 648 (b) 655 (c) 650 (d) 643
3. 3.Which of these numbers is divisible by 3? (a) 1548 (b) 1547 (c) 1549 (d) 1552
4. 4.Which of these numbers is divisible by 11? (a) 2445 (b) 2424 (c) 2431 (d) 2439
5. 5.What is the smallest number you must add to 865 to make it divisible by 12? (a) 11 (b) 1 (c) 9 (d) 10
6. 6.What is the remainder when 410 is divided by 8? (a) 2 (b) 1 (c) 3 (d) 6
7. 7.What is the units digit of 4^24? (a) 6 (b) 4 (c) 7 (d) 5
8. 8.Which of these numbers is divisible by 9? (a) 8116 (b) 8108 (c) 8100 (d) 8095
9. 9.What is the smallest number you must add to 197 to make it divisible by 4? (a) 3 (b) 1 (c) 2
10. 10.What is the smallest number you must add to 120 to make it divisible by 9? (a) 5 (b) 6 (c) 3 (d) 7
11. 11.Which of these numbers is divisible by 4? (a) 837 (b) 833 (c) 842 (d) 836
12. 12.What is the smallest number you must add to 466 to make it divisible by 8? (a) 2 (b) 5 (c) 7 (d) 6
13. 13.What is the units digit of 9^51? (a) 7 (b) 1 (c) 8 (d) 9
14. 14.What is the remainder when 258 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
15. 15.Which of these numbers is divisible by 3? (a) 1387 (b) 1385 (c) 1386 (d) 1390
16. 16.What is the smallest number you must add to 386 to make it divisible by 6? (a) 4 (b) 2 (c) 3 (d) 5
17. 17.Which of these numbers is divisible by 8? (a) 5808 (b) 5823 (c) 5804 (d) 5814
18. 18.What is the remainder when 358 is divided by 6? (a) 2 (b) 5 (c) 4 (d) 3
19. 19.What is the units digit of 7^52? (a) 7 (b) 3 (c) 1 (d) 9
20. 20.Which of these numbers is divisible by 11? (a) 1274 (b) 1254 (c) 1264 (d) 1250
21. 21.Which of these numbers is divisible by 8? (a) 6418 (b) 6411 (c) 6416 (d) 6427
22. 22.Which of these numbers is divisible by 6? (a) 1464 (b) 1463 (c) 1465 (d) 1472
23. 23.What is the smallest number you must add to 248 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
24. 24.Which of these numbers is divisible by 8? (a) 780 (b) 768 (c) 770 (d) 761
25. 25.What is the remainder when 579 is divided by 12? (a) 2 (b) 4 (c) 9 (d) 3
26. 26.What is the units digit of 3^55? (a) 3 (b) 7 (c) 1 (d) 9
27. 27.What is the units digit of 4^20? (a) 6 (b) 7 (c) 4 (d) 5
28. 28.Which of these numbers is divisible by 8? (a) 1716 (b) 1732 (c) 1720 (d) 1722
29. 29.What is the remainder when 140 is divided by 9? (a) 4 (b) 7 (c) 6 (d) 5
30. 30.Which of these numbers is divisible by 8? (a) 1128 (b) 1127 (c) 1134 (d) 1141
31. 31.What is the remainder when 487 is divided by 7? (a) 3 (b) 4 (c) 5 (d) 6
32. 32.What is the remainder when 277 is divided by 6? (a) 2 (b) 0 (c) 5 (d) 1
33. 33.What is the remainder when 186 is divided by 5? (a) 2 (b) 0 (c) 4 (d) 1
34. 34.What is the smallest number you must add to 886 to make it divisible by 11? (a) 5 (b) 6 (c) 4 (d) 7
35. 35.What is the units digit of 8^29? (a) 2 (b) 6 (c) 8 (d) 4
36. 36.What is the remainder when 306 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
37. 37.Which of these numbers is divisible by 11? (a) 5190 (b) 5202 (c) 5176 (d) 5181
38. 38.Which of these numbers is divisible by 6? (a) 2561 (b) 2556 (c) 2551 (d) 2563
39. 39.What is the units digit of 9^57? (a) 1 (b) 7 (c) 9 (d) 8
40. 40.What is the units digit of 4^53? (a) 3 (b) 6 (c) 4 (d) 5

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