Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 780 · 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 4? (a) 2545 (b) 2555 (c) 2551 (d) 2548
2. 2.What is the remainder when 257 is divided by 6? (a) 4 (b) 5 (c) 3 (d) 1
3. 3.What is the smallest number you must add to 318 to make it divisible by 4? (a) 3 (b) 1 (c) 2
4. 4.What is the units digit of 9^34? (a) 1 (b) 0 (c) 2 (d) 9
5. 5.What is the remainder when 243 is divided by 12? (a) 2 (b) 4 (c) 9 (d) 3
6. 6.What is the remainder when 98 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
7. 7.What is the smallest number you must add to 99 to make it divisible by 4? (a) 3 (b) 2 (c) 1
8. 8.What is the smallest number you must add to 221 to make it divisible by 7? (a) 4 (b) 2 (c) 3 (d) 5
9. 9.What is the remainder when 777 is divided by 11? (a) 7 (b) 4 (c) 8 (d) 6
10. 10.What is the units digit of 4^29? (a) 4 (b) 6 (c) 3 (d) 5
11. 11.What is the smallest number you must add to 795 to make it divisible by 11? (a) 7 (b) 9 (c) 8 (d) 3
12. 12.Which of these numbers is divisible by 8? (a) 446 (b) 432 (c) 430 (d) 435
13. 13.What is the smallest number you must add to 86 to make it divisible by 4? (a) 3 (b) 1 (c) 2
14. 14.What is the units digit of 9^55? (a) 9 (b) 1 (c) 7 (d) 8
15. 15.What is the smallest number you must add to 340 to make it divisible by 9? (a) 2 (b) 3 (c) 7 (d) 1
16. 16.What is the units digit of 2^40? (a) 6 (b) 4 (c) 8 (d) 2
17. 17.Which of these numbers is divisible by 4? (a) 1997 (b) 1992 (c) 1990 (d) 1995
18. 18.What is the remainder when 113 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
19. 19.Which of these numbers is divisible by 4? (a) 2912 (b) 2919 (c) 2914 (d) 2910
20. 20.Which of these numbers is divisible by 6? (a) 1398 (b) 1393 (c) 1402 (d) 1405
21. 21.What is the units digit of 4^57? (a) 5 (b) 4 (c) 3 (d) 6
22. 22.What is the units digit of 8^28? (a) 8 (b) 6 (c) 4 (d) 2
23. 23.Which of these numbers is divisible by 11? (a) 3282 (b) 3308 (c) 3289 (d) 3297
24. 24.What is the remainder when 281 is divided by 10? (a) 1 (b) 9 (c) 0 (d) 2
25. 25.What is the smallest number you must add to 726 to make it divisible by 9? (a) 2 (b) 3 (c) 6 (d) 4
26. 26.Which of these numbers is divisible by 11? (a) 3256 (b) 3247 (c) 3268 (d) 3259
27. 27.What is the units digit of 3^24? (a) 9 (b) 7 (c) 3 (d) 1
28. 28.What is the units digit of 8^40? (a) 2 (b) 8 (c) 4 (d) 6
29. 29.What is the units digit of 3^29? (a) 7 (b) 1 (c) 9 (d) 3
30. 30.What is the units digit of 9^43? (a) 9 (b) 7 (c) 8 (d) 1
31. 31.Which of these numbers is divisible by 9? (a) 7988 (b) 7983 (c) 7982 (d) 7996
32. 32.Which of these numbers is divisible by 3? (a) 1813 (b) 1807 (c) 1809 (d) 1811
33. 33.Which of these numbers is divisible by 4? (a) 1137 (b) 1133 (c) 1136 (d) 1141
34. 34.What is the remainder when 271 is divided by 10? (a) 2 (b) 0 (c) 1 (d) 9
35. 35.What is the remainder when 524 is divided by 12? (a) 8 (b) 4 (c) 9 (d) 7
36. 36.What is the units digit of 3^28? (a) 9 (b) 3 (c) 1 (d) 7
37. 37.What is the units digit of 9^45? (a) 8 (b) 7 (c) 9 (d) 1
38. 38.What is the remainder when 468 is divided by 7? (a) 1 (b) 5 (c) 6 (d) 4
39. 39.What is the remainder when 582 is divided by 8? (a) 5 (b) 6 (c) 7 (d) 2
40. 40.Which of these numbers is divisible by 11? (a) 5073 (b) 5056 (c) 5060 (d) 5067

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