Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 569 · 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.Which of these numbers is divisible by 8? (a) 4039 (b) 4044 (c) 4051 (d) 4040
2. 2.What is the units digit of 8^58? (a) 6 (b) 2 (c) 8 (d) 4
3. 3.What is the units digit of 9^22? (a) 1 (b) 2 (c) 9 (d) 0
4. 4.Which of these numbers is divisible by 8? (a) 5508 (b) 5499 (c) 5496 (d) 5494
5. 5.What is the remainder when 307 is divided by 11? (a) 8 (b) 9 (c) 1 (d) 10
6. 6.What is the remainder when 778 is divided by 11? (a) 7 (b) 3 (c) 9 (d) 8
7. 7.Which of these numbers is divisible by 3? (a) 1597 (b) 1596 (c) 1595 (d) 1600
8. 8.What is the remainder when 290 is divided by 8? (a) 6 (b) 1 (c) 2 (d) 3
9. 9.Which of these numbers is divisible by 11? (a) 7288 (b) 7303 (c) 7311 (d) 7293
10. 10.What is the remainder when 107 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
11. 11.What is the units digit of 9^20? (a) 9 (b) 1 (c) 2 (d) 0
12. 12.What is the remainder when 493 is divided by 7? (a) 5 (b) 3 (c) 2 (d) 4
13. 13.Which of these numbers is divisible by 4? (a) 621 (b) 616 (c) 619 (d) 614
14. 14.What is the remainder when 92 is divided by 8? (a) 3 (b) 5 (c) 4 (d) 6
15. 15.What is the remainder when 267 is divided by 7? (a) 6 (b) 0 (c) 1 (d) 2
16. 16.What is the smallest number you must add to 144 to make it divisible by 7? (a) 3 (b) 4 (c) 5 (d) 2
17. 17.What is the remainder when 145 is divided by 10? (a) 6 (b) 4 (c) 5 (d) 7
18. 18.What is the units digit of 3^46? (a) 9 (b) 7 (c) 3 (d) 1
19. 19.Which of these numbers is divisible by 9? (a) 435 (b) 448 (c) 453 (d) 441
20. 20.What is the remainder when 361 is divided by 5? (a) 0 (b) 4 (c) 1 (d) 2
21. 21.What is the smallest number you must add to 247 to make it divisible by 8? (a) 7 (b) 2 (c) 3 (d) 1
22. 22.What is the smallest number you must add to 397 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
23. 23.Which of these numbers is divisible by 4? (a) 3068 (b) 3067 (c) 3074 (d) 3070
24. 24.What is the remainder when 471 is divided by 12? (a) 9 (b) 2 (c) 4 (d) 3
25. 25.What is the smallest number you must add to 689 to make it divisible by 9? (a) 4 (b) 5 (c) 3 (d) 6
26. 26.What is the units digit of 7^33? (a) 7 (b) 3 (c) 1 (d) 9
27. 27.Which of these numbers is divisible by 6? (a) 4947 (b) 4954 (c) 4961 (d) 4950
28. 28.What is the smallest number you must add to 273 to make it divisible by 9? (a) 3 (b) 5 (c) 7 (d) 6
29. 29.What is the smallest number you must add to 522 to make it divisible by 12? (a) 8 (b) 7 (c) 6 (d) 5
30. 30.What is the smallest number you must add to 863 to make it divisible by 11? (a) 5 (b) 6 (c) 8 (d) 7
31. 31.What is the remainder when 54 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
32. 32.Which of these numbers is divisible by 6? (a) 731 (b) 734 (c) 721 (d) 726
33. 33.What is the units digit of 3^26? (a) 1 (b) 7 (c) 9 (d) 3
34. 34.What is the remainder when 718 is divided by 12? (a) 10 (b) 11 (c) 9 (d) 2
35. 35.What is the remainder when 557 is divided by 11? (a) 4 (b) 8 (c) 6 (d) 7
36. 36.What is the units digit of 7^28? (a) 7 (b) 9 (c) 3 (d) 1
37. 37.Which of these numbers is divisible by 8? (a) 3681 (b) 3690 (c) 3688 (d) 3702
38. 38.What is the remainder when 137 is divided by 9? (a) 2 (b) 3 (c) 1 (d) 7
39. 39.What is the units digit of 7^42? (a) 1 (b) 9 (c) 3 (d) 7
40. 40.What is the remainder when 558 is divided by 7? (a) 6 (b) 2 (c) 5 (d) 4

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