Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 598 · 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 remainder when 312 is divided by 9? (a) 3 (b) 6 (c) 5 (d) 7
2. 2.What is the remainder when 642 is divided by 9? (a) 6 (b) 4 (c) 2 (d) 3
3. 3.What is the smallest number you must add to 557 to make it divisible by 8? (a) 3 (b) 2 (c) 5 (d) 4
4. 4.What is the units digit of 2^20? (a) 6 (b) 2 (c) 8 (d) 4
5. 5.What is the smallest number you must add to 218 to make it divisible by 11? (a) 1 (b) 3 (c) 9 (d) 2
6. 6.What is the units digit of 7^30? (a) 3 (b) 1 (c) 7 (d) 9
7. 7.Which of these numbers is divisible by 8? (a) 2841 (b) 2850 (c) 2837 (d) 2840
8. 8.What is the units digit of 3^38? (a) 7 (b) 9 (c) 3 (d) 1
9. 9.What is the remainder when 47 is divided by 3? (a) 1 (b) 0 (c) 2
10. 10.Which of these numbers is divisible by 11? (a) 4334 (b) 4346 (c) 4331 (d) 4336
11. 11.Which of these numbers is divisible by 8? (a) 2764 (b) 2760 (c) 2775 (d) 2754
12. 12.Which of these numbers is divisible by 9? (a) 3705 (b) 3683 (c) 3690 (d) 3695
13. 13.What is the remainder when 317 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
14. 14.What is the smallest number you must add to 741 to make it divisible by 10? (a) 9 (b) 1 (c) 7 (d) 8
15. 15.What is the units digit of 8^55? (a) 2 (b) 4 (c) 8 (d) 6
16. 16.What is the smallest number you must add to 122 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
17. 17.What is the units digit of 3^42? (a) 7 (b) 9 (c) 3 (d) 1
18. 18.What is the smallest number you must add to 449 to make it divisible by 9? (a) 3 (b) 8 (c) 1 (d) 2
19. 19.What is the remainder when 239 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
20. 20.What is the units digit of 8^21? (a) 6 (b) 8 (c) 4 (d) 2
21. 21.What is the units digit of 2^59? (a) 2 (b) 4 (c) 8 (d) 6
22. 22.What is the remainder when 214 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
23. 23.What is the smallest number you must add to 539 to make it divisible by 9? (a) 2 (b) 8 (c) 1 (d) 3
24. 24.What is the smallest number you must add to 237 to make it divisible by 4? (a) 3 (b) 1 (c) 2
25. 25.Which of these numbers is divisible by 8? (a) 6270 (b) 6259 (c) 6256 (d) 6250
26. 26.Which of these numbers is divisible by 3? (a) 2082 (b) 2086 (c) 2080 (d) 2083
27. 27.What is the remainder when 206 is divided by 5? (a) 2 (b) 4 (c) 0 (d) 1
28. 28.Which of these numbers is divisible by 3? (a) 1718 (b) 1721 (c) 1719 (d) 1723
29. 29.What is the remainder when 500 is divided by 7? (a) 4 (b) 2 (c) 5 (d) 3
30. 30.What is the units digit of 2^35? (a) 4 (b) 6 (c) 2 (d) 8
31. 31.Which of these numbers is divisible by 8? (a) 3050 (b) 3047 (c) 3040 (d) 3035
32. 32.What is the units digit of 7^53? (a) 3 (b) 1 (c) 7 (d) 9
33. 33.What is the units digit of 3^45? (a) 9 (b) 1 (c) 7 (d) 3
34. 34.Which of these numbers is divisible by 8? (a) 851 (b) 848 (c) 843 (d) 857
35. 35.What is the units digit of 9^30? (a) 0 (b) 1 (c) 2 (d) 9
36. 36.What is the smallest number you must add to 452 to make it divisible by 11? (a) 1 (b) 9 (c) 10 (d) 8
37. 37.What is the units digit of 2^49? (a) 2 (b) 8 (c) 6 (d) 4
38. 38.What is the units digit of 8^48? (a) 2 (b) 8 (c) 4 (d) 6
39. 39.What is the smallest number you must add to 103 to make it divisible by 3? (a) 1 (b) 2
40. 40.What is the units digit of 8^18? (a) 8 (b) 6 (c) 2 (d) 4

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