Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 30 · 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) 3656 (b) 3649 (c) 3661 (d) 3667
2. 2.What is the smallest number you must add to 355 to make it divisible by 11? (a) 9 (b) 8 (c) 7 (d) 3
3. 3.What is the units digit of 2^49? (a) 8 (b) 6 (c) 4 (d) 2
4. 4.What is the smallest number you must add to 99 to make it divisible by 4? (a) 2 (b) 3 (c) 1
5. 5.What is the remainder when 279 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
6. 6.Which of these numbers is divisible by 3? (a) 2672 (b) 2668 (c) 2670 (d) 2675
7. 7.What is the remainder when 261 is divided by 7? (a) 3 (b) 5 (c) 2 (d) 1
8. 8.Which of these numbers is divisible by 6? (a) 1673 (b) 1657 (c) 1666 (d) 1662
9. 9.Which of these numbers is divisible by 8? (a) 6001 (b) 6009 (c) 6000 (d) 5994
10. 10.What is the units digit of 4^48? (a) 5 (b) 4 (c) 7 (d) 6
11. 11.What is the smallest number you must add to 433 to make it divisible by 11? (a) 7 (b) 4 (c) 8 (d) 6
12. 12.What is the units digit of 4^33? (a) 3 (b) 4 (c) 5 (d) 6
13. 13.Which of these numbers is divisible by 8? (a) 7033 (b) 7032 (c) 7030 (d) 7046
14. 14.What is the units digit of 4^25? (a) 6 (b) 3 (c) 4 (d) 5
15. 15.Which of these numbers is divisible by 9? (a) 5150 (b) 5162 (c) 5157 (d) 5174
16. 16.What is the remainder when 436 is divided by 6? (a) 4 (b) 2 (c) 5 (d) 3
17. 17.What is the units digit of 2^18? (a) 8 (b) 6 (c) 4 (d) 2
18. 18.What is the smallest number you must add to 339 to make it divisible by 10? (a) 3 (b) 2 (c) 1 (d) 9
19. 19.What is the remainder when 283 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
20. 20.What is the remainder when 143 is divided by 6? (a) 3 (b) 1 (c) 5 (d) 4
21. 21.Which of these numbers is divisible by 8? (a) 1707 (b) 1704 (c) 1714 (d) 1701
22. 22.What is the remainder when 173 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
23. 23.What is the smallest number you must add to 233 to make it divisible by 4? (a) 3 (b) 1 (c) 2
24. 24.What is the remainder when 179 is divided by 9? (a) 8 (b) 1 (c) 7 (d) 6
25. 25.What is the units digit of 4^59? (a) 5 (b) 3 (c) 4 (d) 6
26. 26.Which of these numbers is divisible by 3? (a) 1835 (b) 1837 (c) 1836 (d) 1841
27. 27.What is the remainder when 406 is divided by 12? (a) 2 (b) 9 (c) 11 (d) 10
28. 28.Which of these numbers is divisible by 4? (a) 1965 (b) 1968 (c) 1969 (d) 1975
29. 29.Which of these numbers is divisible by 4? (a) 640 (b) 637 (c) 642 (d) 645
30. 30.Which of these numbers is divisible by 3? (a) 2596 (b) 2598 (c) 2600 (d) 2603
31. 31.What is the smallest number you must add to 506 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 2
32. 32.What is the units digit of 2^16? (a) 2 (b) 8 (c) 6 (d) 4
33. 33.What is the units digit of 7^21? (a) 1 (b) 9 (c) 3 (d) 7
34. 34.What is the units digit of 2^34? (a) 8 (b) 6 (c) 2 (d) 4
35. 35.Which of these numbers is divisible by 8? (a) 2330 (b) 2328 (c) 2338 (d) 2326
36. 36.What is the smallest number you must add to 569 to make it divisible by 12? (a) 7 (b) 8 (c) 5 (d) 6
37. 37.Which of these numbers is divisible by 3? (a) 928 (b) 925 (c) 931 (d) 927
38. 38.What is the smallest number you must add to 86 to make it divisible by 3? (a) 2 (b) 1
39. 39.Which of these numbers is divisible by 3? (a) 1088 (b) 1084 (c) 1086 (d) 1091
40. 40.What is the smallest number you must add to 225 to make it divisible by 8? (a) 1 (b) 5 (c) 7 (d) 6

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