Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 815 · 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 units digit of 2^45? (a) 2 (b) 4 (c) 6 (d) 8
2. 2.What is the smallest number you must add to 401 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
3. 3.What is the units digit of 8^34? (a) 2 (b) 6 (c) 8 (d) 4
4. 4.Which of these numbers is divisible by 8? (a) 1943 (b) 1946 (c) 1932 (d) 1936
5. 5.Which of these numbers is divisible by 8? (a) 1268 (b) 1277 (c) 1264 (d) 1261
6. 6.What is the remainder when 103 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
7. 7.What is the units digit of 9^37? (a) 9 (b) 1 (c) 8 (d) 7
8. 8.What is the units digit of 3^34? (a) 1 (b) 3 (c) 7 (d) 9
9. 9.What is the smallest number you must add to 524 to make it divisible by 10? (a) 6 (b) 5 (c) 7 (d) 4
10. 10.What is the units digit of 9^60? (a) 2 (b) 0 (c) 1 (d) 9
11. 11.Which of these numbers is divisible by 8? (a) 5631 (b) 5638 (c) 5617 (d) 5624
12. 12.What is the units digit of 2^22? (a) 2 (b) 8 (c) 6 (d) 4
13. 13.What is the units digit of 3^37? (a) 1 (b) 7 (c) 9 (d) 3
14. 14.What is the units digit of 9^56? (a) 9 (b) 0 (c) 2 (d) 1
15. 15.Which of these numbers is divisible by 3? (a) 2137 (b) 2135 (c) 2131 (d) 2133
16. 16.What is the units digit of 9^28? (a) 2 (b) 0 (c) 9 (d) 1
17. 17.What is the units digit of 3^38? (a) 7 (b) 9 (c) 1 (d) 3
18. 18.Which of these numbers is divisible by 4? (a) 2062 (b) 2065 (c) 2060 (d) 2058
19. 19.Which of these numbers is divisible by 4? (a) 919 (b) 920 (c) 921 (d) 926
20. 20.Which of these numbers is divisible by 4? (a) 1490 (b) 1487 (c) 1484 (d) 1481
21. 21.What is the smallest number you must add to 466 to make it divisible by 6? (a) 3 (b) 2 (c) 4 (d) 1
22. 22.What is the smallest number you must add to 728 to make it divisible by 12? (a) 8 (b) 5 (c) 3 (d) 4
23. 23.What is the units digit of 9^13? (a) 1 (b) 7 (c) 8 (d) 9
24. 24.Which of these numbers is divisible by 6? (a) 2229 (b) 2239 (c) 2232 (d) 2236
25. 25.What is the units digit of 3^47? (a) 9 (b) 3 (c) 1 (d) 7
26. 26.What is the units digit of 7^18? (a) 7 (b) 1 (c) 9 (d) 3
27. 27.What is the smallest number you must add to 574 to make it divisible by 8? (a) 6 (b) 2 (c) 1 (d) 3
28. 28.What is the remainder when 82 is divided by 3? (a) 0 (b) 1 (c) 2
29. 29.Which of these numbers is divisible by 11? (a) 8710 (b) 8717 (c) 8725 (d) 8712
30. 30.Which of these numbers is divisible by 11? (a) 2531 (b) 2527 (c) 2544 (d) 2530
31. 31.What is the units digit of 4^16? (a) 4 (b) 7 (c) 6 (d) 5
32. 32.What is the smallest number you must add to 78 to make it divisible by 4? (a) 1 (b) 3 (c) 2
33. 33.What is the smallest number you must add to 199 to make it divisible by 10? (a) 2 (b) 1 (c) 3 (d) 9
34. 34.What is the smallest number you must add to 34 to make it divisible by 3? (a) 2 (b) 1
35. 35.What is the units digit of 2^54? (a) 8 (b) 4 (c) 6 (d) 2
36. 36.What is the units digit of 9^19? (a) 1 (b) 8 (c) 9 (d) 7
37. 37.Which of these numbers is divisible by 6? (a) 3993 (b) 3988 (c) 3981 (d) 3984
38. 38.What is the smallest number you must add to 167 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
39. 39.What is the remainder when 79 is divided by 3? (a) 2 (b) 1 (c) 0
40. 40.Which of these numbers is divisible by 9? (a) 5893 (b) 5900 (c) 5886 (d) 5878

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