Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 808 · 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 145 to make it divisible by 4? (a) 2 (b) 1 (c) 3
2. 2.What is the remainder when 764 is divided by 10? (a) 6 (b) 5 (c) 4 (d) 3
3. 3.Which of these numbers is divisible by 3? (a) 757 (b) 763 (c) 760 (d) 759
4. 4.What is the remainder when 173 is divided by 3? (a) 1 (b) 2 (c) 0
5. 5.What is the remainder when 940 is divided by 12? (a) 8 (b) 4 (c) 5 (d) 3
6. 6.Which of these numbers is divisible by 6? (a) 777 (b) 790 (c) 780 (d) 782
7. 7.What is the units digit of 8^37? (a) 4 (b) 2 (c) 8 (d) 6
8. 8.What is the remainder when 53 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
9. 9.What is the units digit of 8^20? (a) 2 (b) 8 (c) 6 (d) 4
10. 10.What is the units digit of 7^50? (a) 1 (b) 7 (c) 9 (d) 3
11. 11.What is the units digit of 4^37? (a) 6 (b) 5 (c) 3 (d) 4
12. 12.What is the smallest number you must add to 158 to make it divisible by 12? (a) 11 (b) 9 (c) 2 (d) 10
13. 13.Which of these numbers is divisible by 8? (a) 5442 (b) 5439 (c) 5449 (d) 5440
14. 14.What is the remainder when 123 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
15. 15.What is the smallest number you must add to 123 to make it divisible by 8? (a) 5 (b) 3 (c) 6 (d) 4
16. 16.What is the smallest number you must add to 411 to make it divisible by 6? (a) 2 (b) 4 (c) 5 (d) 3
17. 17.What is the remainder when 302 is divided by 12? (a) 1 (b) 3 (c) 2 (d) 10
18. 18.What is the remainder when 456 is divided by 10? (a) 7 (b) 4 (c) 6 (d) 5
19. 19.Which of these numbers is divisible by 4? (a) 511 (b) 503 (c) 504 (d) 507
20. 20.What is the remainder when 93 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
21. 21.What is the units digit of 8^23? (a) 4 (b) 6 (c) 8 (d) 2
22. 22.What is the remainder when 177 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
23. 23.Which of these numbers is divisible by 6? (a) 4234 (b) 4239 (c) 4244 (d) 4236
24. 24.What is the units digit of 8^28? (a) 6 (b) 4 (c) 8 (d) 2
25. 25.What is the units digit of 7^53? (a) 3 (b) 9 (c) 7 (d) 1
26. 26.Which of these numbers is divisible by 11? (a) 5148 (b) 5145 (c) 5158 (d) 5166
27. 27.What is the smallest number you must add to 295 to make it divisible by 4? (a) 1 (b) 3 (c) 2
28. 28.Which of these numbers is divisible by 4? (a) 1309 (b) 1305 (c) 1308 (d) 1314
29. 29.What is the units digit of 4^35? (a) 6 (b) 3 (c) 5 (d) 4
30. 30.What is the remainder when 289 is divided by 10? (a) 7 (b) 8 (c) 1 (d) 9
31. 31.What is the units digit of 4^27? (a) 4 (b) 5 (c) 3 (d) 6
32. 32.Which of these numbers is divisible by 4? (a) 687 (b) 682 (c) 684 (d) 689
33. 33.What is the smallest number you must add to 188 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
34. 34.What is the smallest number you must add to 87 to make it divisible by 4? (a) 3 (b) 2 (c) 1
35. 35.What is the units digit of 9^18? (a) 1 (b) 2 (c) 0 (d) 9
36. 36.What is the units digit of 3^60? (a) 7 (b) 9 (c) 3 (d) 1
37. 37.What is the units digit of 3^37? (a) 9 (b) 3 (c) 1 (d) 7
38. 38.Which of these numbers is divisible by 8? (a) 7072 (b) 7077 (c) 7083 (d) 7066
39. 39.What is the remainder when 419 is divided by 10? (a) 9 (b) 8 (c) 7 (d) 1
40. 40.Which of these numbers is divisible by 6? (a) 2933 (b) 2934 (c) 2943 (d) 2935

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