Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 580 · 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 2^18? (a) 8 (b) 2 (c) 6 (d) 4
2. 2.What is the units digit of 3^29? (a) 3 (b) 1 (c) 7 (d) 9
3. 3.What is the units digit of 2^24? (a) 2 (b) 4 (c) 8 (d) 6
4. 4.Which of these numbers is divisible by 9? (a) 7607 (b) 7618 (c) 7605 (d) 7600
5. 5.What is the remainder when 346 is divided by 5? (a) 2 (b) 0 (c) 4 (d) 1
6. 6.What is the units digit of 4^27? (a) 5 (b) 3 (c) 4 (d) 6
7. 7.Which of these numbers is divisible by 3? (a) 1123 (b) 1120 (c) 1126 (d) 1122
8. 8.Which of these numbers is divisible by 3? (a) 186 (b) 190 (c) 188 (d) 185
9. 9.What is the smallest number you must add to 35 to make it divisible by 3? (a) 1 (b) 2
10. 10.What is the units digit of 7^11? (a) 9 (b) 1 (c) 3 (d) 7
11. 11.What is the remainder when 671 is divided by 10? (a) 9 (b) 1 (c) 2 (d) 0
12. 12.What is the smallest number you must add to 334 to make it divisible by 9? (a) 6 (b) 8 (c) 7 (d) 1
13. 13.Which of these numbers is divisible by 4? (a) 3005 (b) 3001 (c) 2999 (d) 3000
14. 14.Which of these numbers is divisible by 6? (a) 853 (b) 849 (c) 852 (d) 859
15. 15.What is the units digit of 3^52? (a) 7 (b) 3 (c) 1 (d) 9
16. 16.Which of these numbers is divisible by 11? (a) 5538 (b) 5556 (c) 5547 (d) 5544
17. 17.What is the remainder when 196 is divided by 3? (a) 0 (b) 1 (c) 2
18. 18.Which of these numbers is divisible by 3? (a) 853 (b) 855 (c) 860 (d) 856
19. 19.What is the units digit of 4^38? (a) 6 (b) 5 (c) 7 (d) 4
20. 20.Which of these numbers is divisible by 4? (a) 1862 (b) 1853 (c) 1858 (d) 1856
21. 21.What is the smallest number you must add to 377 to make it divisible by 11? (a) 7 (b) 9 (c) 8 (d) 3
22. 22.What is the smallest number you must add to 284 to make it divisible by 7? (a) 5 (b) 3 (c) 4 (d) 2
23. 23.Which of these numbers is divisible by 6? (a) 4782 (b) 4778 (c) 4783 (d) 4793
24. 24.What is the smallest number you must add to 601 to make it divisible by 8? (a) 5 (b) 6 (c) 1 (d) 7
25. 25.What is the units digit of 9^52? (a) 0 (b) 1 (c) 9 (d) 2
26. 26.What is the units digit of 9^28? (a) 0 (b) 9 (c) 2 (d) 1
27. 27.What is the remainder when 501 is divided by 11? (a) 5 (b) 6 (c) 8 (d) 7
28. 28.What is the remainder when 478 is divided by 11? (a) 6 (b) 7 (c) 5 (d) 4
29. 29.What is the smallest number you must add to 452 to make it divisible by 11? (a) 8 (b) 9 (c) 1 (d) 10
30. 30.What is the units digit of 7^14? (a) 3 (b) 1 (c) 7 (d) 9
31. 31.Which of these numbers is divisible by 11? (a) 5094 (b) 5104 (c) 5118 (d) 5110
32. 32.Which of these numbers is divisible by 6? (a) 2232 (b) 2237 (c) 2243 (d) 2227
33. 33.What is the units digit of 3^10? (a) 1 (b) 9 (c) 3 (d) 7
34. 34.What is the units digit of 3^20? (a) 9 (b) 1 (c) 7 (d) 3
35. 35.Which of these numbers is divisible by 6? (a) 1436 (b) 1429 (c) 1428 (d) 1427
36. 36.What is the smallest number you must add to 203 to make it divisible by 3? (a) 2 (b) 1
37. 37.What is the remainder when 294 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
38. 38.What is the smallest number you must add to 85 to make it divisible by 7? (a) 5 (b) 4 (c) 1 (d) 6
39. 39.What is the remainder when 74 is divided by 3? (a) 0 (b) 2 (c) 1
40. 40.What is the smallest number you must add to 677 to make it divisible by 9? (a) 8 (b) 7 (c) 2 (d) 6

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