Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 939 · 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 796 to make it divisible by 10? (a) 6 (b) 4 (c) 3 (d) 5
2. 2.What is the remainder when 768 is divided by 10? (a) 7 (b) 8 (c) 2 (d) 9
3. 3.What is the remainder when 317 is divided by 7? (a) 3 (b) 1 (c) 5 (d) 2
4. 4.What is the remainder when 206 is divided by 3? (a) 0 (b) 2 (c) 1
5. 5.What is the smallest number you must add to 162 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 1
6. 6.Which of these numbers is divisible by 3? (a) 2240 (b) 2236 (c) 2238 (d) 2242
7. 7.What is the remainder when 375 is divided by 10? (a) 7 (b) 4 (c) 5 (d) 6
8. 8.What is the units digit of 3^19? (a) 9 (b) 3 (c) 7 (d) 1
9. 9.What is the remainder when 737 is divided by 12? (a) 4 (b) 5 (c) 6 (d) 7
10. 10.What is the remainder when 258 is divided by 8? (a) 2 (b) 1 (c) 6 (d) 3
11. 11.Which of these numbers is divisible by 9? (a) 596 (b) 615 (c) 603 (d) 608
12. 12.What is the remainder when 230 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
13. 13.What is the units digit of 3^33? (a) 1 (b) 7 (c) 9 (d) 3
14. 14.Which of these numbers is divisible by 6? (a) 4254 (b) 4256 (c) 4250 (d) 4261
15. 15.What is the units digit of 7^26? (a) 9 (b) 7 (c) 1 (d) 3
16. 16.What is the remainder when 516 is divided by 9? (a) 6 (b) 3 (c) 4 (d) 2
17. 17.What is the remainder when 431 is divided by 12? (a) 10 (b) 11 (c) 9 (d) 1
18. 18.What is the units digit of 7^60? (a) 1 (b) 3 (c) 9 (d) 7
19. 19.Which of these numbers is divisible by 3? (a) 1839 (b) 1844 (c) 1837 (d) 1840
20. 20.What is the remainder when 256 is divided by 7? (a) 3 (b) 6 (c) 4 (d) 5
21. 21.What is the units digit of 7^14? (a) 1 (b) 7 (c) 3 (d) 9
22. 22.Which of these numbers is divisible by 3? (a) 754 (b) 749 (c) 752 (d) 750
23. 23.What is the units digit of 4^37? (a) 3 (b) 4 (c) 6 (d) 5
24. 24.What is the units digit of 2^34? (a) 4 (b) 8 (c) 2 (d) 6
25. 25.What is the remainder when 226 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
26. 26.What is the remainder when 324 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
27. 27.Which of these numbers is divisible by 6? (a) 4399 (b) 4404 (c) 4413 (d) 4407
28. 28.What is the remainder when 416 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 2
29. 29.Which of these numbers is divisible by 3? (a) 301 (b) 305 (c) 303 (d) 307
30. 30.What is the units digit of 8^27? (a) 2 (b) 6 (c) 4 (d) 8
31. 31.What is the remainder when 536 is divided by 9? (a) 4 (b) 5 (c) 7 (d) 6
32. 32.What is the units digit of 4^13? (a) 5 (b) 3 (c) 6 (d) 4
33. 33.What is the smallest number you must add to 73 to make it divisible by 3? (a) 1 (b) 2
34. 34.Which of these numbers is divisible by 6? (a) 3367 (b) 3364 (c) 3375 (d) 3366
35. 35.What is the remainder when 41 is divided by 3? (a) 2 (b) 0 (c) 1
36. 36.What is the remainder when 602 is divided by 9? (a) 7 (b) 8 (c) 6 (d) 1
37. 37.Which of these numbers is divisible by 6? (a) 3879 (b) 3882 (c) 3885 (d) 3893
38. 38.Which of these numbers is divisible by 11? (a) 715 (b) 728 (c) 713 (d) 718
39. 39.What is the smallest number you must add to 178 to make it divisible by 6? (a) 1 (b) 2 (c) 4 (d) 3
40. 40.What is the remainder when 662 is divided by 12? (a) 10 (b) 2 (c) 1 (d) 3

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