Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 299 · 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 199 to make it divisible by 8? (a) 2 (b) 7 (c) 1 (d) 3
2. 2.What is the units digit of 8^25? (a) 2 (b) 6 (c) 4 (d) 8
3. 3.What is the remainder when 311 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
4. 4.What is the smallest number you must add to 338 to make it divisible by 11? (a) 8 (b) 2 (c) 3 (d) 4
5. 5.What is the units digit of 9^56? (a) 0 (b) 1 (c) 9 (d) 2
6. 6.Which of these numbers is divisible by 3? (a) 1763 (b) 1769 (c) 1764 (d) 1765
7. 7.What is the units digit of 2^15? (a) 8 (b) 6 (c) 4 (d) 2
8. 8.What is the remainder when 377 is divided by 8? (a) 7 (b) 0 (c) 1 (d) 2
9. 9.Which of these numbers is divisible by 9? (a) 2281 (b) 2287 (c) 2296 (d) 2286
10. 10.Which of these numbers is divisible by 8? (a) 715 (b) 711 (c) 704 (d) 702
11. 11.Which of these numbers is divisible by 6? (a) 694 (b) 700 (c) 707 (d) 696
12. 12.What is the units digit of 4^11? (a) 4 (b) 3 (c) 5 (d) 6
13. 13.What is the smallest number you must add to 134 to make it divisible by 11? (a) 10 (b) 8 (c) 2 (d) 9
14. 14.What is the smallest number you must add to 545 to make it divisible by 12? (a) 6 (b) 8 (c) 7 (d) 5
15. 15.What is the units digit of 4^57? (a) 4 (b) 5 (c) 6 (d) 3
16. 16.What is the smallest number you must add to 534 to make it divisible by 10? (a) 6 (b) 5 (c) 7 (d) 4
17. 17.Which of these numbers is divisible by 9? (a) 7400 (b) 7392 (c) 7386 (d) 7389
18. 18.What is the remainder when 686 is divided by 11? (a) 4 (b) 7 (c) 3 (d) 5
19. 19.What is the remainder when 242 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
20. 20.What is the smallest number you must add to 289 to make it divisible by 9? (a) 8 (b) 7 (c) 6 (d) 1
21. 21.What is the remainder when 41 is divided by 3? (a) 0 (b) 2 (c) 1
22. 22.What is the units digit of 8^15? (a) 8 (b) 4 (c) 6 (d) 2
23. 23.What is the smallest number you must add to 387 to make it divisible by 6? (a) 5 (b) 4 (c) 2 (d) 3
24. 24.Which of these numbers is divisible by 3? (a) 615 (b) 616 (c) 620 (d) 614
25. 25.Which of these numbers is divisible by 4? (a) 574 (b) 583 (c) 577 (d) 576
26. 26.Which of these numbers is divisible by 6? (a) 4704 (b) 4712 (c) 4700 (d) 4708
27. 27.Which of these numbers is divisible by 6? (a) 2591 (b) 2596 (c) 2584 (d) 2586
28. 28.What is the remainder when 129 is divided by 6? (a) 4 (b) 5 (c) 2 (d) 3
29. 29.What is the units digit of 3^49? (a) 9 (b) 1 (c) 3 (d) 7
30. 30.What is the smallest number you must add to 95 to make it divisible by 4? (a) 2 (b) 3 (c) 1
31. 31.What is the units digit of 7^33? (a) 1 (b) 3 (c) 9 (d) 7
32. 32.What is the smallest number you must add to 322 to make it divisible by 4? (a) 3 (b) 2 (c) 1
33. 33.Which of these numbers is divisible by 6? (a) 4389 (b) 4381 (c) 4386 (d) 4395
34. 34.What is the remainder when 205 is divided by 10? (a) 7 (b) 6 (c) 5 (d) 4
35. 35.Which of these numbers is divisible by 3? (a) 1176 (b) 1181 (c) 1178 (d) 1175
36. 36.What is the units digit of 8^10? (a) 6 (b) 4 (c) 8 (d) 2
37. 37.What is the remainder when 440 is divided by 6? (a) 3 (b) 4 (c) 1 (d) 2
38. 38.What is the units digit of 9^55? (a) 1 (b) 9 (c) 7 (d) 8
39. 39.What is the remainder when 100 is divided by 6? (a) 2 (b) 4 (c) 3 (d) 5
40. 40.Which of these numbers is divisible by 8? (a) 2084 (b) 2091 (c) 2080 (d) 2073

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