Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 555 · 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 470 to make it divisible by 12? (a) 11 (b) 2 (c) 10 (d) 9
2. 2.What is the remainder when 52 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
3. 3.Which of these numbers is divisible by 8? (a) 2725 (b) 2715 (c) 2712 (d) 2707
4. 4.Which of these numbers is divisible by 3? (a) 227 (b) 228 (c) 233 (d) 230
5. 5.Which of these numbers is divisible by 6? (a) 4374 (b) 4376 (c) 4382 (d) 4373
6. 6.What is the remainder when 740 is divided by 11? (a) 3 (b) 8 (c) 4 (d) 2
7. 7.What is the remainder when 324 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
8. 8.What is the smallest number you must add to 34 to make it divisible by 3? (a) 2 (b) 1
9. 9.Which of these numbers is divisible by 3? (a) 1553 (b) 1549 (c) 1548 (d) 1547
10. 10.What is the smallest number you must add to 302 to make it divisible by 9? (a) 4 (b) 3 (c) 5 (d) 6
11. 11.Which of these numbers is divisible by 6? (a) 2659 (b) 2665 (c) 2673 (d) 2664
12. 12.What is the units digit of 8^20? (a) 8 (b) 4 (c) 2 (d) 6
13. 13.What is the units digit of 9^58? (a) 0 (b) 1 (c) 2 (d) 9
14. 14.Which of these numbers is divisible by 8? (a) 1127 (b) 1131 (c) 1118 (d) 1120
15. 15.What is the smallest number you must add to 268 to make it divisible by 8? (a) 6 (b) 4 (c) 3 (d) 5
16. 16.What is the smallest number you must add to 93 to make it divisible by 4? (a) 1 (b) 2 (c) 3
17. 17.What is the smallest number you must add to 827 to make it divisible by 11? (a) 10 (b) 9 (c) 8 (d) 2
18. 18.What is the smallest number you must add to 627 to make it divisible by 10? (a) 4 (b) 2 (c) 7 (d) 3
19. 19.What is the units digit of 7^18? (a) 7 (b) 9 (c) 1 (d) 3
20. 20.What is the smallest number you must add to 134 to make it divisible by 3? (a) 1 (b) 2
21. 21.What is the units digit of 4^19? (a) 4 (b) 3 (c) 5 (d) 6
22. 22.What is the units digit of 9^47? (a) 1 (b) 8 (c) 7 (d) 9
23. 23.Which of these numbers is divisible by 11? (a) 4537 (b) 4563 (c) 4548 (d) 4543
24. 24.Which of these numbers is divisible by 11? (a) 5115 (b) 5117 (c) 5114 (d) 5129
25. 25.What is the remainder when 615 is divided by 9? (a) 2 (b) 4 (c) 6 (d) 3
26. 26.What is the remainder when 148 is divided by 3? (a) 0 (b) 1 (c) 2
27. 27.Which of these numbers is divisible by 6? (a) 1949 (b) 1943 (c) 1944 (d) 1954
28. 28.What is the smallest number you must add to 385 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 1
29. 29.Which of these numbers is divisible by 8? (a) 2777 (b) 2768 (c) 2764 (d) 2772
30. 30.Which of these numbers is divisible by 3? (a) 2038 (b) 2044 (c) 2040 (d) 2042
31. 31.What is the smallest number you must add to 337 to make it divisible by 12? (a) 11 (b) 1 (c) 9 (d) 10
32. 32.What is the units digit of 2^23? (a) 8 (b) 4 (c) 2 (d) 6
33. 33.What is the units digit of 8^37? (a) 4 (b) 8 (c) 2 (d) 6
34. 34.What is the smallest number you must add to 516 to make it divisible by 7? (a) 5 (b) 3 (c) 1 (d) 2
35. 35.Which of these numbers is divisible by 9? (a) 1828 (b) 1839 (c) 1836 (d) 1847
36. 36.Which of these numbers is divisible by 4? (a) 1003 (b) 999 (c) 1000 (d) 1005
37. 37.Which of these numbers is divisible by 4? (a) 349 (b) 352 (c) 357 (d) 353
38. 38.What is the units digit of 8^57? (a) 4 (b) 8 (c) 2 (d) 6
39. 39.What is the smallest number you must add to 164 to make it divisible by 7? (a) 3 (b) 4 (c) 5 (d) 6
40. 40.What is the units digit of 8^34? (a) 2 (b) 4 (c) 8 (d) 6

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