Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 830 · 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) 4337 (b) 4331 (c) 4328 (d) 4322
2. 2.What is the smallest number you must add to 147 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
3. 3.Which of these numbers is divisible by 11? (a) 2532 (b) 2543 (c) 2554 (d) 2541
4. 4.What is the units digit of 4^37? (a) 3 (b) 4 (c) 5 (d) 6
5. 5.Which of these numbers is divisible by 8? (a) 1858 (b) 1867 (c) 1864 (d) 1873
6. 6.Which of these numbers is divisible by 6? (a) 3190 (b) 3196 (c) 3192 (d) 3199
7. 7.What is the remainder when 98 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
8. 8.What is the smallest number you must add to 202 to make it divisible by 3? (a) 2 (b) 1
9. 9.What is the units digit of 4^22? (a) 7 (b) 5 (c) 6 (d) 4
10. 10.What is the remainder when 402 is divided by 8? (a) 1 (b) 2 (c) 3 (d) 6
11. 11.Which of these numbers is divisible by 9? (a) 2197 (b) 2218 (c) 2205 (d) 2211
12. 12.What is the remainder when 590 is divided by 11? (a) 6 (b) 7 (c) 8 (d) 4
13. 13.What is the smallest number you must add to 187 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 1
14. 14.Which of these numbers is divisible by 8? (a) 6101 (b) 6094 (c) 6086 (d) 6088
15. 15.What is the units digit of 2^10? (a) 6 (b) 8 (c) 4 (d) 2
16. 16.What is the smallest number you must add to 686 to make it divisible by 11? (a) 8 (b) 4 (c) 7 (d) 6
17. 17.What is the units digit of 7^22? (a) 7 (b) 9 (c) 1 (d) 3
18. 18.What is the remainder when 742 is divided by 10? (a) 8 (b) 3 (c) 2 (d) 1
19. 19.What is the smallest number you must add to 765 to make it divisible by 11? (a) 7 (b) 4 (c) 5 (d) 6
20. 20.Which of these numbers is divisible by 4? (a) 1762 (b) 1765 (c) 1759 (d) 1760
21. 21.What is the smallest number you must add to 129 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
22. 22.What is the remainder when 832 is divided by 11? (a) 8 (b) 6 (c) 4 (d) 7
23. 23.Which of these numbers is divisible by 9? (a) 1510 (b) 1512 (c) 1527 (d) 1517
24. 24.What is the units digit of 4^56? (a) 6 (b) 4 (c) 7 (d) 5
25. 25.Which of these numbers is divisible by 3? (a) 470 (b) 476 (c) 472 (d) 471
26. 26.What is the remainder when 926 is divided by 12? (a) 3 (b) 10 (c) 1 (d) 2
27. 27.What is the remainder when 505 is divided by 9? (a) 0 (b) 2 (c) 8 (d) 1
28. 28.Which of these numbers is divisible by 9? (a) 5657 (b) 5638 (c) 5643 (d) 5650
29. 29.What is the smallest number you must add to 591 to make it divisible by 9? (a) 2 (b) 6 (c) 4 (d) 3
30. 30.What is the remainder when 739 is divided by 12? (a) 5 (b) 8 (c) 7 (d) 6
31. 31.What is the remainder when 67 is divided by 3? (a) 2 (b) 0 (c) 1
32. 32.Which of these numbers is divisible by 4? (a) 1579 (b) 1572 (c) 1571 (d) 1575
33. 33.What is the smallest number you must add to 446 to make it divisible by 10? (a) 4 (b) 6 (c) 5 (d) 3
34. 34.Which of these numbers is divisible by 3? (a) 1539 (b) 1541 (c) 1544 (d) 1537
35. 35.Which of these numbers is divisible by 11? (a) 3364 (b) 3384 (c) 3366 (d) 3369
36. 36.What is the units digit of 4^28? (a) 7 (b) 4 (c) 6 (d) 5
37. 37.Which of these numbers is divisible by 3? (a) 1868 (b) 1863 (c) 1865 (d) 1861
38. 38.Which of these numbers is divisible by 6? (a) 1177 (b) 1172 (c) 1176 (d) 1185
39. 39.What is the units digit of 2^32? (a) 2 (b) 4 (c) 8 (d) 6
40. 40.Which of these numbers is divisible by 8? (a) 4325 (b) 4319 (c) 4320 (d) 4335

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