Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 456 · 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 242 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
2. 2.Which of these numbers is divisible by 3? (a) 1942 (b) 1941 (c) 1939 (d) 1946
3. 3.Which of these numbers is divisible by 4? (a) 1183 (b) 1176 (c) 1175 (d) 1178
4. 4.What is the smallest number you must add to 185 to make it divisible by 10? (a) 4 (b) 6 (c) 7 (d) 5
5. 5.What is the remainder when 118 is divided by 3? (a) 1 (b) 0 (c) 2
6. 6.What is the smallest number you must add to 299 to make it divisible by 9? (a) 6 (b) 8 (c) 7 (d) 2
7. 7.What is the smallest number you must add to 259 to make it divisible by 11? (a) 6 (b) 7 (c) 5 (d) 4
8. 8.Which of these numbers is divisible by 4? (a) 305 (b) 301 (c) 300 (d) 298
9. 9.What is the smallest number you must add to 97 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 1
10. 10.What is the remainder when 181 is divided by 3? (a) 1 (b) 2 (c) 0
11. 11.What is the units digit of 8^23? (a) 8 (b) 6 (c) 4 (d) 2
12. 12.What is the smallest number you must add to 143 to make it divisible by 4? (a) 1 (b) 2 (c) 3
13. 13.What is the units digit of 9^46? (a) 9 (b) 0 (c) 2 (d) 1
14. 14.Which of these numbers is divisible by 9? (a) 2616 (b) 2624 (c) 2609 (d) 2610
15. 15.What is the remainder when 223 is divided by 9? (a) 8 (b) 7 (c) 6 (d) 2
16. 16.What is the smallest number you must add to 790 to make it divisible by 12? (a) 3 (b) 10 (c) 2 (d) 1
17. 17.What is the remainder when 713 is divided by 10? (a) 2 (b) 7 (c) 3 (d) 4
18. 18.What is the remainder when 289 is divided by 6? (a) 1 (b) 2 (c) 0 (d) 5
19. 19.What is the remainder when 283 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
20. 20.Which of these numbers is divisible by 6? (a) 3182 (b) 3179 (c) 3189 (d) 3180
21. 21.What is the units digit of 7^38? (a) 7 (b) 3 (c) 9 (d) 1
22. 22.Which of these numbers is divisible by 6? (a) 3551 (b) 3554 (c) 3552 (d) 3562
23. 23.What is the smallest number you must add to 164 to make it divisible by 3? (a) 1 (b) 2
24. 24.Which of these numbers is divisible by 11? (a) 1004 (b) 996 (c) 1001 (d) 1016
25. 25.What is the units digit of 8^51? (a) 8 (b) 4 (c) 6 (d) 2
26. 26.What is the units digit of 8^20? (a) 2 (b) 4 (c) 6 (d) 8
27. 27.Which of these numbers is divisible by 9? (a) 4852 (b) 4842 (c) 4839 (d) 4843
28. 28.Which of these numbers is divisible by 8? (a) 4446 (b) 4448 (c) 4453 (d) 4458
29. 29.What is the remainder when 213 is divided by 11? (a) 3 (b) 5 (c) 4 (d) 7
30. 30.What is the smallest number you must add to 193 to make it divisible by 4? (a) 3 (b) 1 (c) 2
31. 31.What is the units digit of 3^28? (a) 7 (b) 1 (c) 3 (d) 9
32. 32.What is the smallest number you must add to 484 to make it divisible by 10? (a) 4 (b) 5 (c) 6 (d) 7
33. 33.What is the smallest number you must add to 387 to make it divisible by 8? (a) 6 (b) 5 (c) 4 (d) 3
34. 34.What is the smallest number you must add to 89 to make it divisible by 4? (a) 1 (b) 2 (c) 3
35. 35.Which of these numbers is divisible by 3? (a) 776 (b) 777 (c) 779 (d) 782
36. 36.What is the remainder when 200 is divided by 7? (a) 3 (b) 5 (c) 4 (d) 6
37. 37.What is the remainder when 190 is divided by 7? (a) 2 (b) 1 (c) 6 (d) 0
38. 38.What is the remainder when 601 is divided by 8? (a) 7 (b) 0 (c) 2 (d) 1
39. 39.What is the units digit of 4^34? (a) 7 (b) 4 (c) 5 (d) 6
40. 40.What is the remainder when 551 is divided by 7? (a) 2 (b) 5 (c) 4 (d) 6

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