Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 744 · 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 smallest number you must add to 384 to make it divisible by 7? (a) 6 (b) 3 (c) 1 (d) 2
2. 2.What is the remainder when 402 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
3. 3.Which of these numbers is divisible by 9? (a) 2216 (b) 2210 (c) 2227 (d) 2214
4. 4.Which of these numbers is divisible by 6? (a) 5082 (b) 5077 (c) 5083 (d) 5092
5. 5.Which of these numbers is divisible by 9? (a) 1798 (b) 1815 (c) 1804 (d) 1800
6. 6.Which of these numbers is divisible by 3? (a) 1145 (b) 1150 (c) 1147 (d) 1146
7. 7.What is the units digit of 8^31? (a) 6 (b) 4 (c) 2 (d) 8
8. 8.Which of these numbers is divisible by 3? (a) 504 (b) 509 (c) 505 (d) 503
9. 9.Which of these numbers is divisible by 6? (a) 5200 (b) 5194 (c) 5196 (d) 5205
10. 10.What is the smallest number you must add to 191 to make it divisible by 3? (a) 1 (b) 2
11. 11.What is the smallest number you must add to 197 to make it divisible by 3? (a) 2 (b) 1
12. 12.Which of these numbers is divisible by 4? (a) 1693 (b) 1688 (c) 1690 (d) 1685
13. 13.What is the units digit of 2^48? (a) 6 (b) 4 (c) 8 (d) 2
14. 14.What is the units digit of 2^41? (a) 8 (b) 4 (c) 6 (d) 2
15. 15.Which of these numbers is divisible by 6? (a) 1183 (b) 1188 (c) 1190 (d) 1196
16. 16.What is the smallest number you must add to 193 to make it divisible by 7? (a) 5 (b) 4 (c) 2 (d) 3
17. 17.What is the smallest number you must add to 83 to make it divisible by 8? (a) 4 (b) 3 (c) 5 (d) 6
18. 18.Which of these numbers is divisible by 3? (a) 1965 (b) 1966 (c) 1963 (d) 1969
19. 19.What is the units digit of 4^12? (a) 7 (b) 6 (c) 5 (d) 4
20. 20.What is the remainder when 146 is divided by 3? (a) 2 (b) 1 (c) 0
21. 21.What is the remainder when 199 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
22. 22.What is the smallest number you must add to 899 to make it divisible by 12? (a) 11 (b) 2 (c) 3 (d) 1
23. 23.What is the units digit of 2^24? (a) 6 (b) 2 (c) 4 (d) 8
24. 24.What is the remainder when 147 is divided by 8? (a) 3 (b) 5 (c) 4 (d) 2
25. 25.What is the units digit of 4^44? (a) 6 (b) 5 (c) 7 (d) 4
26. 26.Which of these numbers is divisible by 4? (a) 1800 (b) 1803 (c) 1797 (d) 1805
27. 27.What is the units digit of 3^11? (a) 1 (b) 7 (c) 3 (d) 9
28. 28.What is the units digit of 9^45? (a) 7 (b) 8 (c) 9 (d) 1
29. 29.Which of these numbers is divisible by 8? (a) 5781 (b) 5770 (c) 5766 (d) 5768
30. 30.What is the units digit of 9^11? (a) 8 (b) 1 (c) 7 (d) 9
31. 31.What is the remainder when 719 is divided by 11? (a) 4 (b) 5 (c) 7 (d) 3
32. 32.Which of these numbers is divisible by 11? (a) 4768 (b) 4774 (c) 4789 (d) 4783
33. 33.What is the smallest number you must add to 560 to make it divisible by 12? (a) 3 (b) 4 (c) 5 (d) 8
34. 34.What is the remainder when 289 is divided by 6? (a) 5 (b) 2 (c) 1 (d) 0
35. 35.What is the smallest number you must add to 571 to make it divisible by 8? (a) 4 (b) 5 (c) 3 (d) 6
36. 36.What is the smallest number you must add to 55 to make it divisible by 3? (a) 2 (b) 1
37. 37.What is the units digit of 8^59? (a) 6 (b) 4 (c) 8 (d) 2
38. 38.Which of these numbers is divisible by 9? (a) 1350 (b) 1361 (c) 1358 (d) 1345
39. 39.Which of these numbers is divisible by 11? (a) 9132 (b) 9110 (c) 9119 (d) 9123
40. 40.Which of these numbers is divisible by 6? (a) 3781 (b) 3796 (c) 3786 (d) 3788

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