Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 343 · 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 55 to make it divisible by 3? (a) 1 (b) 2
2. 2.What is the units digit of 8^15? (a) 2 (b) 8 (c) 4 (d) 6
3. 3.What is the remainder when 149 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
4. 4.What is the smallest number you must add to 205 to make it divisible by 3? (a) 2 (b) 1
5. 5.What is the units digit of 4^15? (a) 3 (b) 4 (c) 5 (d) 6
6. 6.What is the smallest number you must add to 41 to make it divisible by 4? (a) 3 (b) 2 (c) 1
7. 7.What is the smallest number you must add to 499 to make it divisible by 9? (a) 4 (b) 6 (c) 5 (d) 7
8. 8.Which of these numbers is divisible by 11? (a) 7678 (b) 7698 (c) 7681 (d) 7673
9. 9.What is the units digit of 8^53? (a) 8 (b) 6 (c) 2 (d) 4
10. 10.What is the units digit of 4^23? (a) 3 (b) 5 (c) 4 (d) 6
11. 11.What is the units digit of 9^36? (a) 0 (b) 9 (c) 2 (d) 1
12. 12.What is the units digit of 7^47? (a) 7 (b) 9 (c) 3 (d) 1
13. 13.Which of these numbers is divisible by 11? (a) 9534 (b) 9557 (c) 9537 (d) 9541
14. 14.What is the units digit of 9^49? (a) 8 (b) 9 (c) 1 (d) 7
15. 15.What is the smallest number you must add to 101 to make it divisible by 4? (a) 1 (b) 2 (c) 3
16. 16.Which of these numbers is divisible by 3? (a) 1978 (b) 1981 (c) 1975 (d) 1977
17. 17.What is the units digit of 4^31? (a) 3 (b) 5 (c) 6 (d) 4
18. 18.What is the units digit of 3^27? (a) 9 (b) 1 (c) 3 (d) 7
19. 19.Which of these numbers is divisible by 3? (a) 1451 (b) 1453 (c) 1448 (d) 1449
20. 20.Which of these numbers is divisible by 11? (a) 3496 (b) 3477 (c) 3487 (d) 3500
21. 21.What is the smallest number you must add to 464 to make it divisible by 12? (a) 4 (b) 8 (c) 5 (d) 3
22. 22.What is the remainder when 206 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
23. 23.Which of these numbers is divisible by 9? (a) 482 (b) 473 (c) 477 (d) 493
24. 24.What is the remainder when 361 is divided by 6? (a) 5 (b) 2 (c) 0 (d) 1
25. 25.What is the units digit of 8^52? (a) 2 (b) 6 (c) 8 (d) 4
26. 26.Which of these numbers is divisible by 11? (a) 4183 (b) 4206 (c) 4194 (d) 4191
27. 27.What is the remainder when 329 is divided by 10? (a) 8 (b) 1 (c) 7 (d) 9
28. 28.What is the units digit of 7^12? (a) 9 (b) 3 (c) 1 (d) 7
29. 29.What is the remainder when 308 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 2
30. 30.Which of these numbers is divisible by 4? (a) 1189 (b) 1199 (c) 1192 (d) 1195
31. 31.What is the units digit of 3^16? (a) 3 (b) 9 (c) 7 (d) 1
32. 32.What is the remainder when 106 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
33. 33.What is the units digit of 9^25? (a) 1 (b) 7 (c) 8 (d) 9
34. 34.What is the smallest number you must add to 286 to make it divisible by 7? (a) 1 (b) 3 (c) 6 (d) 2
35. 35.What is the smallest number you must add to 434 to make it divisible by 12? (a) 11 (b) 9 (c) 2 (d) 10
36. 36.Which of these numbers is divisible by 3? (a) 2015 (b) 2018 (c) 2021 (d) 2016
37. 37.What is the remainder when 329 is divided by 12? (a) 4 (b) 5 (c) 6 (d) 7
38. 38.Which of these numbers is divisible by 6? (a) 2377 (b) 2370 (c) 2375 (d) 2366
39. 39.What is the remainder when 442 is divided by 6? (a) 3 (b) 5 (c) 2 (d) 4
40. 40.What is the units digit of 4^37? (a) 5 (b) 4 (c) 6 (d) 3

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