Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 465 · 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 9? (a) 1990 (b) 1981 (c) 1980 (d) 1973
2. 2.What is the remainder when 145 is divided by 9? (a) 2 (b) 8 (c) 0 (d) 1
3. 3.What is the remainder when 651 is divided by 10? (a) 9 (b) 1 (c) 0 (d) 2
4. 4.What is the units digit of 2^13? (a) 6 (b) 8 (c) 2 (d) 4
5. 5.What is the remainder when 161 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
6. 6.What is the smallest number you must add to 187 to make it divisible by 3? (a) 1 (b) 2
7. 7.What is the units digit of 4^12? (a) 5 (b) 7 (c) 6 (d) 4
8. 8.Which of these numbers is divisible by 8? (a) 6990 (b) 6978 (c) 6999 (d) 6984
9. 9.What is the remainder when 153 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
10. 10.What is the units digit of 7^26? (a) 1 (b) 3 (c) 9 (d) 7
11. 11.Which of these numbers is divisible by 9? (a) 3481 (b) 3487 (c) 3483 (d) 3496
12. 12.Which of these numbers is divisible by 9? (a) 2718 (b) 2713 (c) 2734 (d) 2725
13. 13.What is the units digit of 9^21? (a) 1 (b) 9 (c) 8 (d) 7
14. 14.What is the units digit of 7^14? (a) 1 (b) 7 (c) 9 (d) 3
15. 15.What is the remainder when 387 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
16. 16.What is the units digit of 9^17? (a) 1 (b) 9 (c) 8 (d) 7
17. 17.What is the smallest number you must add to 190 to make it divisible by 11? (a) 8 (b) 3 (c) 7 (d) 9
18. 18.Which of these numbers is divisible by 9? (a) 1580 (b) 1567 (c) 1585 (d) 1575
19. 19.What is the smallest number you must add to 295 to make it divisible by 4? (a) 1 (b) 3 (c) 2
20. 20.What is the units digit of 8^16? (a) 6 (b) 4 (c) 2 (d) 8
21. 21.What is the smallest number you must add to 93 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
22. 22.What is the remainder when 224 is divided by 3? (a) 2 (b) 0 (c) 1
23. 23.What is the smallest number you must add to 656 to make it divisible by 11? (a) 4 (b) 5 (c) 3 (d) 7
24. 24.Which of these numbers is divisible by 3? (a) 2312 (b) 2309 (c) 2307 (d) 2306
25. 25.What is the smallest number you must add to 486 to make it divisible by 11? (a) 10 (b) 8 (c) 9 (d) 2
26. 26.What is the remainder when 811 is divided by 11? (a) 9 (b) 8 (c) 7 (d) 3
27. 27.What is the remainder when 100 is divided by 7? (a) 5 (b) 3 (c) 2 (d) 1
28. 28.What is the units digit of 9^47? (a) 7 (b) 9 (c) 8 (d) 1
29. 29.What is the remainder when 306 is divided by 10? (a) 5 (b) 6 (c) 7 (d) 4
30. 30.Which of these numbers is divisible by 6? (a) 4789 (b) 4796 (c) 4788 (d) 4787
31. 31.Which of these numbers is divisible by 11? (a) 8899 (b) 8911 (c) 8896 (d) 8903
32. 32.What is the remainder when 371 is divided by 5? (a) 0 (b) 2 (c) 4 (d) 1
33. 33.What is the remainder when 193 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
34. 34.What is the remainder when 178 is divided by 7? (a) 4 (b) 3 (c) 2 (d) 5
35. 35.Which of these numbers is divisible by 3? (a) 1094 (b) 1096 (c) 1095 (d) 1099
36. 36.Which of these numbers is divisible by 6? (a) 5082 (b) 5081 (c) 5093 (d) 5083
37. 37.What is the smallest number you must add to 776 to make it divisible by 12? (a) 4 (b) 3 (c) 5 (d) 8
38. 38.What is the smallest number you must add to 98 to make it divisible by 9? (a) 8 (b) 3 (c) 2 (d) 1
39. 39.What is the smallest number you must add to 479 to make it divisible by 6? (a) 5 (b) 3 (c) 2 (d) 1
40. 40.Which of these numbers is divisible by 4? (a) 963 (b) 965 (c) 960 (d) 957

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