Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 578 · 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 4^59? (a) 5 (b) 6 (c) 3 (d) 4
2. 2.What is the remainder when 439 is divided by 6? (a) 1 (b) 0 (c) 2 (d) 5
3. 3.Which of these numbers is divisible by 4? (a) 1972 (b) 1970 (c) 1975 (d) 1979
4. 4.What is the units digit of 4^12? (a) 5 (b) 7 (c) 4 (d) 6
5. 5.What is the units digit of 2^16? (a) 6 (b) 8 (c) 4 (d) 2
6. 6.Which of these numbers is divisible by 6? (a) 1150 (b) 1153 (c) 1162 (d) 1152
7. 7.What is the remainder when 56 is divided by 3? (a) 2 (b) 1 (c) 0
8. 8.What is the smallest number you must add to 746 to make it divisible by 12? (a) 2 (b) 10 (c) 9 (d) 11
9. 9.What is the remainder when 931 is divided by 12? (a) 5 (b) 7 (c) 8 (d) 6
10. 10.What is the units digit of 3^11? (a) 9 (b) 1 (c) 3 (d) 7
11. 11.Which of these numbers is divisible by 8? (a) 1637 (b) 1646 (c) 1655 (d) 1640
12. 12.Which of these numbers is divisible by 3? (a) 2463 (b) 2462 (c) 2467 (d) 2465
13. 13.What is the remainder when 165 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
14. 14.Which of these numbers is divisible by 8? (a) 2393 (b) 2379 (c) 2386 (d) 2384
15. 15.Which of these numbers is divisible by 9? (a) 5436 (b) 5442 (c) 5453 (d) 5432
16. 16.What is the remainder when 636 is divided by 8? (a) 5 (b) 3 (c) 6 (d) 4
17. 17.What is the smallest number you must add to 578 to make it divisible by 11? (a) 4 (b) 5 (c) 7 (d) 6
18. 18.Which of these numbers is divisible by 4? (a) 775 (b) 783 (c) 776 (d) 777
19. 19.What is the remainder when 88 is divided by 3? (a) 2 (b) 1 (c) 0
20. 20.What is the remainder when 57 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
21. 21.What is the remainder when 222 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
22. 22.What is the units digit of 2^57? (a) 6 (b) 2 (c) 8 (d) 4
23. 23.Which of these numbers is divisible by 6? (a) 2079 (b) 2070 (c) 2071 (d) 2065
24. 24.What is the units digit of 2^24? (a) 6 (b) 2 (c) 8 (d) 4
25. 25.What is the units digit of 9^49? (a) 8 (b) 7 (c) 9 (d) 1
26. 26.Which of these numbers is divisible by 8? (a) 3681 (b) 3690 (c) 3688 (d) 3701
27. 27.Which of these numbers is divisible by 11? (a) 8261 (b) 8264 (c) 8277 (d) 8258
28. 28.What is the smallest number you must add to 342 to make it divisible by 10? (a) 8 (b) 7 (c) 2 (d) 9
29. 29.What is the units digit of 2^32? (a) 6 (b) 4 (c) 2 (d) 8
30. 30.What is the units digit of 4^49? (a) 5 (b) 6 (c) 4 (d) 3
31. 31.Which of these numbers is divisible by 11? (a) 7122 (b) 7109 (c) 7106 (d) 7099
32. 32.What is the units digit of 2^12? (a) 6 (b) 4 (c) 2 (d) 8
33. 33.Which of these numbers is divisible by 4? (a) 1059 (b) 1062 (c) 1054 (d) 1056
34. 34.Which of these numbers is divisible by 4? (a) 3051 (b) 3045 (c) 3053 (d) 3048
35. 35.Which of these numbers is divisible by 6? (a) 2894 (b) 2903 (c) 2892 (d) 2891
36. 36.What is the remainder when 142 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
37. 37.Which of these numbers is divisible by 8? (a) 2053 (b) 2063 (c) 2056 (d) 2066
38. 38.What is the remainder when 565 is divided by 9? (a) 7 (b) 2 (c) 6 (d) 8
39. 39.What is the units digit of 4^26? (a) 5 (b) 6 (c) 4 (d) 7
40. 40.What is the smallest number you must add to 524 to make it divisible by 9? (a) 7 (b) 2 (c) 8 (d) 6

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