Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 518 · 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 8? (a) 1114 (b) 1122 (c) 1107 (d) 1112
2. 2.Which of these numbers is divisible by 3? (a) 2119 (b) 2123 (c) 2121 (d) 2125
3. 3.Which of these numbers is divisible by 9? (a) 544 (b) 540 (c) 550 (d) 538
4. 4.What is the remainder when 123 is divided by 10? (a) 7 (b) 2 (c) 4 (d) 3
5. 5.What is the remainder when 147 is divided by 6? (a) 5 (b) 3 (c) 4 (d) 2
6. 6.What is the units digit of 2^19? (a) 2 (b) 6 (c) 8 (d) 4
7. 7.What is the remainder when 272 is divided by 6? (a) 3 (b) 2 (c) 1 (d) 4
8. 8.What is the remainder when 132 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
9. 9.What is the smallest number you must add to 417 to make it divisible by 8? (a) 7 (b) 1 (c) 5 (d) 6
10. 10.What is the remainder when 637 is divided by 11? (a) 10 (b) 9 (c) 1 (d) 8
11. 11.Which of these numbers is divisible by 9? (a) 5458 (b) 5445 (c) 5437 (d) 5448
12. 12.What is the units digit of 8^15? (a) 4 (b) 8 (c) 6 (d) 2
13. 13.Which of these numbers is divisible by 8? (a) 966 (b) 978 (c) 968 (d) 974
14. 14.Which of these numbers is divisible by 3? (a) 1892 (b) 1895 (c) 1889 (d) 1890
15. 15.Which of these numbers is divisible by 6? (a) 2423 (b) 2418 (c) 2417 (d) 2429
16. 16.What is the smallest number you must add to 202 to make it divisible by 10? (a) 7 (b) 8 (c) 2 (d) 9
17. 17.What is the remainder when 203 is divided by 3? (a) 1 (b) 0 (c) 2
18. 18.Which of these numbers is divisible by 9? (a) 2456 (b) 2462 (c) 2457 (d) 2467
19. 19.Which of these numbers is divisible by 8? (a) 334 (b) 323 (c) 342 (d) 328
20. 20.What is the smallest number you must add to 302 to make it divisible by 7? (a) 4 (b) 6 (c) 1 (d) 5
21. 21.What is the remainder when 215 is divided by 11? (a) 7 (b) 8 (c) 6 (d) 5
22. 22.Which of these numbers is divisible by 4? (a) 2415 (b) 2409 (c) 2412 (d) 2417
23. 23.What is the remainder when 62 is divided by 3? (a) 0 (b) 2 (c) 1
24. 24.What is the smallest number you must add to 674 to make it divisible by 10? (a) 7 (b) 4 (c) 6 (d) 5
25. 25.What is the units digit of 9^45? (a) 7 (b) 9 (c) 8 (d) 1
26. 26.What is the units digit of 2^33? (a) 8 (b) 6 (c) 2 (d) 4
27. 27.What is the remainder when 435 is divided by 9? (a) 6 (b) 3 (c) 2 (d) 4
28. 28.What is the smallest number you must add to 387 to make it divisible by 6? (a) 5 (b) 2 (c) 4 (d) 3
29. 29.Which of these numbers is divisible by 4? (a) 2538 (b) 2542 (c) 2536 (d) 2533
30. 30.What is the remainder when 125 is divided by 6? (a) 5 (b) 1 (c) 3 (d) 4
31. 31.What is the units digit of 7^42? (a) 3 (b) 1 (c) 9 (d) 7
32. 32.What is the remainder when 308 is divided by 5? (a) 2 (b) 1 (c) 3 (d) 4
33. 33.What is the units digit of 4^54? (a) 7 (b) 6 (c) 4 (d) 5
34. 34.What is the smallest number you must add to 456 to make it divisible by 11? (a) 5 (b) 8 (c) 7 (d) 6
35. 35.What is the smallest number you must add to 433 to make it divisible by 10? (a) 6 (b) 8 (c) 7 (d) 3
36. 36.What is the smallest number you must add to 241 to make it divisible by 3? (a) 1 (b) 2
37. 37.What is the smallest number you must add to 318 to make it divisible by 4? (a) 3 (b) 2 (c) 1
38. 38.What is the units digit of 7^56? (a) 7 (b) 3 (c) 9 (d) 1
39. 39.Which of these numbers is divisible by 11? (a) 7023 (b) 7029 (c) 7032 (d) 7050
40. 40.What is the smallest number you must add to 113 to make it divisible by 4? (a) 1 (b) 2 (c) 3

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