Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 452 · 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 615 to make it divisible by 9? (a) 7 (b) 5 (c) 3 (d) 6
2. 2.What is the remainder when 131 is divided by 3? (a) 2 (b) 0 (c) 1
3. 3.What is the units digit of 4^10? (a) 5 (b) 4 (c) 6 (d) 7
4. 4.Which of these numbers is divisible by 3? (a) 655 (b) 654 (c) 658 (d) 653
5. 5.Which of these numbers is divisible by 11? (a) 1732 (b) 1716 (c) 1711 (d) 1719
6. 6.What is the smallest number you must add to 671 to make it divisible by 12? (a) 2 (b) 3 (c) 11 (d) 1
7. 7.Which of these numbers is divisible by 6? (a) 409 (b) 405 (c) 419 (d) 408
8. 8.What is the remainder when 281 is divided by 10? (a) 2 (b) 0 (c) 1 (d) 9
9. 9.What is the smallest number you must add to 298 to make it divisible by 12? (a) 10 (b) 3 (c) 1 (d) 2
10. 10.What is the remainder when 636 is divided by 8? (a) 6 (b) 5 (c) 3 (d) 4
11. 11.What is the smallest number you must add to 354 to make it divisible by 10? (a) 6 (b) 4 (c) 5 (d) 7
12. 12.Which of these numbers is divisible by 9? (a) 3829 (b) 3824 (c) 3825 (d) 3840
13. 13.What is the units digit of 8^19? (a) 8 (b) 4 (c) 2 (d) 6
14. 14.What is the units digit of 4^51? (a) 4 (b) 3 (c) 5 (d) 6
15. 15.What is the smallest number you must add to 461 to make it divisible by 12? (a) 7 (b) 6 (c) 5 (d) 8
16. 16.Which of these numbers is divisible by 4? (a) 2996 (b) 3001 (c) 2998 (d) 2994
17. 17.What is the remainder when 141 is divided by 7? (a) 0 (b) 2 (c) 6 (d) 1
18. 18.Which of these numbers is divisible by 8? (a) 5076 (b) 5062 (c) 5068 (d) 5064
19. 19.What is the remainder when 284 is divided by 6? (a) 1 (b) 2 (c) 4 (d) 3
20. 20.What is the units digit of 3^45? (a) 7 (b) 3 (c) 1 (d) 9
21. 21.Which of these numbers is divisible by 3? (a) 1264 (b) 1259 (c) 1260 (d) 1262
22. 22.What is the smallest number you must add to 362 to make it divisible by 8? (a) 2 (b) 5 (c) 7 (d) 6
23. 23.Which of these numbers is divisible by 4? (a) 2332 (b) 2330 (c) 2338 (d) 2334
24. 24.What is the units digit of 9^18? (a) 9 (b) 0 (c) 2 (d) 1
25. 25.What is the remainder when 504 is divided by 11? (a) 9 (b) 10 (c) 8 (d) 2
26. 26.What is the smallest number you must add to 565 to make it divisible by 8? (a) 4 (b) 3 (c) 2 (d) 5
27. 27.What is the smallest number you must add to 267 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
28. 28.What is the units digit of 7^18? (a) 1 (b) 9 (c) 3 (d) 7
29. 29.What is the remainder when 560 is divided by 9? (a) 1 (b) 2 (c) 7 (d) 3
30. 30.What is the smallest number you must add to 297 to make it divisible by 4? (a) 3 (b) 2 (c) 1
31. 31.Which of these numbers is divisible by 11? (a) 8894 (b) 8869 (c) 8879 (d) 8877
32. 32.Which of these numbers is divisible by 3? (a) 2348 (b) 2342 (c) 2343 (d) 2345
33. 33.What is the remainder when 102 is divided by 8? (a) 5 (b) 6 (c) 2 (d) 7
34. 34.What is the remainder when 476 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 2
35. 35.Which of these numbers is divisible by 4? (a) 2678 (b) 2685 (c) 2680 (d) 2683
36. 36.Which of these numbers is divisible by 9? (a) 1475 (b) 1476 (c) 1490 (d) 1477
37. 37.Which of these numbers is divisible by 6? (a) 4034 (b) 4032 (c) 4028 (d) 4040
38. 38.Which of these numbers is divisible by 11? (a) 1290 (b) 1282 (c) 1276 (d) 1270
39. 39.What is the smallest number you must add to 573 to make it divisible by 11? (a) 10 (b) 8 (c) 1 (d) 9
40. 40.What is the units digit of 8^26? (a) 2 (b) 8 (c) 6 (d) 4

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