Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 243 · 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 4? (a) 3333 (b) 3343 (c) 3337 (d) 3336
2. 2.What is the units digit of 8^36? (a) 6 (b) 4 (c) 2 (d) 8
3. 3.What is the smallest number you must add to 106 to make it divisible by 10? (a) 3 (b) 5 (c) 6 (d) 4
4. 4.What is the smallest number you must add to 288 to make it divisible by 10? (a) 3 (b) 2 (c) 1 (d) 8
5. 5.What is the smallest number you must add to 591 to make it divisible by 8? (a) 7 (b) 2 (c) 3 (d) 1
6. 6.Which of these numbers is divisible by 9? (a) 937 (b) 927 (c) 934 (d) 920
7. 7.What is the units digit of 3^17? (a) 9 (b) 1 (c) 7 (d) 3
8. 8.Which of these numbers is divisible by 11? (a) 4461 (b) 4466 (c) 4482 (d) 4476
9. 9.What is the smallest number you must add to 272 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 2
10. 10.What is the units digit of 2^12? (a) 4 (b) 8 (c) 2 (d) 6
11. 11.What is the units digit of 8^22? (a) 8 (b) 4 (c) 6 (d) 2
12. 12.What is the remainder when 137 is divided by 6? (a) 3 (b) 4 (c) 5 (d) 1
13. 13.Which of these numbers is divisible by 9? (a) 3201 (b) 3186 (c) 3193 (d) 3180
14. 14.What is the units digit of 3^42? (a) 3 (b) 1 (c) 7 (d) 9
15. 15.Which of these numbers is divisible by 6? (a) 515 (b) 507 (c) 510 (d) 520
16. 16.Which of these numbers is divisible by 4? (a) 1608 (b) 1607 (c) 1613 (d) 1610
17. 17.Which of these numbers is divisible by 8? (a) 1158 (b) 1160 (c) 1174 (d) 1164
18. 18.What is the units digit of 9^49? (a) 8 (b) 1 (c) 9 (d) 7
19. 19.Which of these numbers is divisible by 4? (a) 1691 (b) 1694 (c) 1692 (d) 1698
20. 20.What is the smallest number you must add to 145 to make it divisible by 3? (a) 1 (b) 2
21. 21.Which of these numbers is divisible by 6? (a) 3012 (b) 3008 (c) 3017 (d) 3023
22. 22.What is the smallest number you must add to 220 to make it divisible by 3? (a) 1 (b) 2
23. 23.What is the units digit of 8^55? (a) 8 (b) 4 (c) 2 (d) 6
24. 24.What is the smallest number you must add to 402 to make it divisible by 9? (a) 4 (b) 2 (c) 3 (d) 6
25. 25.What is the units digit of 8^54? (a) 2 (b) 6 (c) 8 (d) 4
26. 26.Which of these numbers is divisible by 11? (a) 7853 (b) 7864 (c) 7854 (d) 7870
27. 27.What is the units digit of 3^44? (a) 3 (b) 9 (c) 7 (d) 1
28. 28.Which of these numbers is divisible by 6? (a) 280 (b) 266 (c) 270 (d) 271
29. 29.Which of these numbers is divisible by 3? (a) 1057 (b) 1055 (c) 1056 (d) 1060
30. 30.Which of these numbers is divisible by 3? (a) 2176 (b) 2170 (c) 2174 (d) 2172
31. 31.What is the remainder when 101 is divided by 3? (a) 2 (b) 0 (c) 1
32. 32.What is the units digit of 2^39? (a) 4 (b) 8 (c) 6 (d) 2
33. 33.What is the units digit of 4^47? (a) 4 (b) 3 (c) 6 (d) 5
34. 34.Which of these numbers is divisible by 3? (a) 1760 (b) 1757 (c) 1755 (d) 1753
35. 35.What is the smallest number you must add to 473 to make it divisible by 10? (a) 7 (b) 6 (c) 8 (d) 3
36. 36.What is the units digit of 4^44? (a) 5 (b) 4 (c) 7 (d) 6
37. 37.What is the smallest number you must add to 57 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
38. 38.What is the remainder when 121 is divided by 3? (a) 1 (b) 2 (c) 0
39. 39.What is the remainder when 291 is divided by 5? (a) 2 (b) 0 (c) 4 (d) 1
40. 40.Which of these numbers is divisible by 8? (a) 5690 (b) 5686 (c) 5701 (d) 5688

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