Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 674 · 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 213 to make it divisible by 12? (a) 3 (b) 4 (c) 2 (d) 9
2. 2.What is the remainder when 120 is divided by 9? (a) 4 (b) 6 (c) 3 (d) 2
3. 3.What is the smallest number you must add to 106 to make it divisible by 3? (a) 1 (b) 2
4. 4.What is the smallest number you must add to 329 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
5. 5.Which of these numbers is divisible by 9? (a) 2876 (b) 2862 (c) 2858 (d) 2865
6. 6.Which of these numbers is divisible by 3? (a) 2410 (b) 2407 (c) 2406 (d) 2405
7. 7.What is the remainder when 236 is divided by 3? (a) 1 (b) 0 (c) 2
8. 8.What is the smallest number you must add to 196 to make it divisible by 3? (a) 1 (b) 2
9. 9.What is the remainder when 125 is divided by 9? (a) 6 (b) 1 (c) 8 (d) 7
10. 10.What is the units digit of 7^31? (a) 9 (b) 3 (c) 7 (d) 1
11. 11.What is the smallest number you must add to 297 to make it divisible by 4? (a) 2 (b) 3 (c) 1
12. 12.What is the units digit of 9^59? (a) 7 (b) 1 (c) 8 (d) 9
13. 13.What is the units digit of 3^15? (a) 7 (b) 3 (c) 1 (d) 9
14. 14.Which of these numbers is divisible by 11? (a) 2566 (b) 2560 (c) 2545 (d) 2552
15. 15.What is the smallest number you must add to 294 to make it divisible by 11? (a) 3 (b) 4 (c) 2 (d) 8
16. 16.What is the remainder when 141 is divided by 5? (a) 0 (b) 1 (c) 2 (d) 4
17. 17.Which of these numbers is divisible by 11? (a) 2167 (b) 2171 (c) 2160 (d) 2179
18. 18.What is the remainder when 113 is divided by 3? (a) 1 (b) 0 (c) 2
19. 19.What is the units digit of 8^35? (a) 6 (b) 4 (c) 2 (d) 8
20. 20.Which of these numbers is divisible by 3? (a) 1640 (b) 1642 (c) 1636 (d) 1638
21. 21.What is the remainder when 134 is divided by 9? (a) 6 (b) 7 (c) 8 (d) 1
22. 22.What is the units digit of 4^34? (a) 5 (b) 7 (c) 6 (d) 4
23. 23.What is the remainder when 337 is divided by 7? (a) 0 (b) 6 (c) 1 (d) 2
24. 24.Which of these numbers is divisible by 6? (a) 2388 (b) 2398 (c) 2393 (d) 2384
25. 25.What is the units digit of 3^60? (a) 3 (b) 9 (c) 7 (d) 1
26. 26.What is the units digit of 7^54? (a) 1 (b) 7 (c) 3 (d) 9
27. 27.What is the remainder when 201 is divided by 5? (a) 2 (b) 4 (c) 0 (d) 1
28. 28.Which of these numbers is divisible by 6? (a) 1052 (b) 1045 (c) 1040 (d) 1044
29. 29.Which of these numbers is divisible by 9? (a) 3221 (b) 3203 (c) 3204 (d) 3206
30. 30.What is the smallest number you must add to 347 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
31. 31.What is the smallest number you must add to 323 to make it divisible by 10? (a) 8 (b) 6 (c) 3 (d) 7
32. 32.What is the units digit of 9^18? (a) 9 (b) 1 (c) 0 (d) 2
33. 33.What is the smallest number you must add to 264 to make it divisible by 9? (a) 6 (b) 3 (c) 5 (d) 7
34. 34.Which of these numbers is divisible by 3? (a) 2644 (b) 2642 (c) 2639 (d) 2640
35. 35.Which of these numbers is divisible by 6? (a) 864 (b) 872 (c) 869 (d) 859
36. 36.What is the smallest number you must add to 607 to make it divisible by 12? (a) 7 (b) 4 (c) 6 (d) 5
37. 37.Which of these numbers is divisible by 11? (a) 1111 (b) 1119 (c) 1107 (d) 1129
38. 38.What is the remainder when 529 is divided by 7? (a) 4 (b) 5 (c) 6 (d) 3
39. 39.Which of these numbers is divisible by 8? (a) 3914 (b) 3904 (c) 3901 (d) 3911
40. 40.What is the remainder when 765 is divided by 11? (a) 6 (b) 8 (c) 7 (d) 5

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