Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 346 · 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 2^10? (a) 8 (b) 2 (c) 6 (d) 4
2. 2.What is the remainder when 523 is divided by 9? (a) 2 (b) 1 (c) 8 (d) 0
3. 3.What is the units digit of 7^50? (a) 7 (b) 3 (c) 9 (d) 1
4. 4.Which of these numbers is divisible by 9? (a) 1072 (b) 1055 (c) 1062 (d) 1064
5. 5.What is the remainder when 261 is divided by 6? (a) 4 (b) 5 (c) 2 (d) 3
6. 6.Which of these numbers is divisible by 3? (a) 1061 (b) 1054 (c) 1056 (d) 1058
7. 7.What is the smallest number you must add to 229 to make it divisible by 7? (a) 1 (b) 5 (c) 3 (d) 2
8. 8.What is the units digit of 7^53? (a) 1 (b) 7 (c) 9 (d) 3
9. 9.Which of these numbers is divisible by 4? (a) 897 (b) 893 (c) 891 (d) 892
10. 10.What is the smallest number you must add to 430 to make it divisible by 6? (a) 4 (b) 2 (c) 1 (d) 3
11. 11.What is the units digit of 2^26? (a) 4 (b) 8 (c) 6 (d) 2
12. 12.What is the units digit of 4^13? (a) 4 (b) 6 (c) 3 (d) 5
13. 13.What is the smallest number you must add to 185 to make it divisible by 3? (a) 2 (b) 1
14. 14.Which of these numbers is divisible by 11? (a) 1192 (b) 1199 (c) 1217 (d) 1204
15. 15.Which of these numbers is divisible by 8? (a) 5232 (b) 5245 (c) 5225 (d) 5237
16. 16.Which of these numbers is divisible by 3? (a) 2630 (b) 2631 (c) 2635 (d) 2632
17. 17.What is the units digit of 7^25? (a) 7 (b) 9 (c) 1 (d) 3
18. 18.Which of these numbers is divisible by 9? (a) 562 (b) 573 (c) 577 (d) 567
19. 19.What is the units digit of 8^60? (a) 2 (b) 8 (c) 4 (d) 6
20. 20.What is the smallest number you must add to 110 to make it divisible by 4? (a) 1 (b) 2 (c) 3
21. 21.What is the units digit of 4^40? (a) 4 (b) 7 (c) 5 (d) 6
22. 22.What is the remainder when 509 is divided by 11? (a) 8 (b) 4 (c) 3 (d) 2
23. 23.What is the smallest number you must add to 569 to make it divisible by 11? (a) 4 (b) 2 (c) 8 (d) 3
24. 24.What is the smallest number you must add to 257 to make it divisible by 12? (a) 5 (b) 7 (c) 6 (d) 8
25. 25.What is the units digit of 3^15? (a) 9 (b) 3 (c) 7 (d) 1
26. 26.Which of these numbers is divisible by 6? (a) 3627 (b) 3619 (c) 3634 (d) 3624
27. 27.What is the smallest number you must add to 359 to make it divisible by 10? (a) 2 (b) 1 (c) 9 (d) 3
28. 28.Which of these numbers is divisible by 8? (a) 4767 (b) 4760 (c) 4753 (d) 4774
29. 29.Which of these numbers is divisible by 11? (a) 9686 (b) 9691 (c) 9711 (d) 9699
30. 30.What is the units digit of 2^54? (a) 4 (b) 2 (c) 8 (d) 6
31. 31.What is the units digit of 9^59? (a) 7 (b) 8 (c) 9 (d) 1
32. 32.What is the smallest number you must add to 183 to make it divisible by 12? (a) 9 (b) 10 (c) 8 (d) 3
33. 33.What is the smallest number you must add to 98 to make it divisible by 6? (a) 4 (b) 2 (c) 5 (d) 3
34. 34.What is the remainder when 599 is divided by 10? (a) 8 (b) 9 (c) 7 (d) 1
35. 35.Which of these numbers is divisible by 6? (a) 791 (b) 797 (c) 803 (d) 792
36. 36.Which of these numbers is divisible by 3? (a) 1429 (b) 1436 (c) 1432 (d) 1431
37. 37.What is the units digit of 7^21? (a) 1 (b) 9 (c) 7 (d) 3
38. 38.Which of these numbers is divisible by 4? (a) 2741 (b) 2736 (c) 2734 (d) 2739
39. 39.What is the units digit of 8^18? (a) 6 (b) 2 (c) 4 (d) 8
40. 40.What is the smallest number you must add to 147 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2

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