Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 647 · 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 6? (a) 4632 (b) 4640 (c) 4637 (d) 4629
2. 2.What is the smallest number you must add to 215 to make it divisible by 12? (a) 1 (b) 2 (c) 11 (d) 3
3. 3.What is the units digit of 8^51? (a) 2 (b) 4 (c) 8 (d) 6
4. 4.Which of these numbers is divisible by 11? (a) 7348 (b) 7351 (c) 7361 (d) 7345
5. 5.What is the remainder when 132 is divided by 10? (a) 1 (b) 2 (c) 3 (d) 8
6. 6.Which of these numbers is divisible by 4? (a) 185 (b) 182 (c) 180 (d) 177
7. 7.What is the units digit of 7^41? (a) 9 (b) 1 (c) 3 (d) 7
8. 8.What is the remainder when 177 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
9. 9.What is the remainder when 253 is divided by 6? (a) 5 (b) 0 (c) 2 (d) 1
10. 10.What is the smallest number you must add to 107 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
11. 11.What is the remainder when 144 is divided by 7? (a) 4 (b) 3 (c) 6 (d) 5
12. 12.What is the units digit of 8^26? (a) 2 (b) 8 (c) 6 (d) 4
13. 13.What is the remainder when 623 is divided by 10? (a) 3 (b) 2 (c) 4 (d) 7
14. 14.What is the units digit of 8^18? (a) 6 (b) 8 (c) 2 (d) 4
15. 15.Which of these numbers is divisible by 4? (a) 3311 (b) 3307 (c) 3315 (d) 3308
16. 16.Which of these numbers is divisible by 9? (a) 5708 (b) 5729 (c) 5715 (d) 5718
17. 17.What is the units digit of 9^46? (a) 0 (b) 9 (c) 2 (d) 1
18. 18.What is the remainder when 761 is divided by 11? (a) 9 (b) 2 (c) 1 (d) 3
19. 19.Which of these numbers is divisible by 3? (a) 1713 (b) 1714 (c) 1718 (d) 1711
20. 20.What is the remainder when 353 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
21. 21.What is the smallest number you must add to 71 to make it divisible by 3? (a) 2 (b) 1
22. 22.What is the remainder when 354 is divided by 9? (a) 2 (b) 6 (c) 3 (d) 4
23. 23.What is the remainder when 433 is divided by 10? (a) 2 (b) 3 (c) 7 (d) 4
24. 24.What is the smallest number you must add to 262 to make it divisible by 12? (a) 1 (b) 3 (c) 2 (d) 10
25. 25.What is the units digit of 4^25? (a) 4 (b) 3 (c) 6 (d) 5
26. 26.What is the units digit of 7^37? (a) 9 (b) 7 (c) 3 (d) 1
27. 27.Which of these numbers is divisible by 4? (a) 1481 (b) 1480 (c) 1477 (d) 1485
28. 28.What is the remainder when 215 is divided by 8? (a) 6 (b) 5 (c) 1 (d) 7
29. 29.What is the units digit of 3^36? (a) 9 (b) 1 (c) 7 (d) 3
30. 30.What is the remainder when 307 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
31. 31.What is the smallest number you must add to 725 to make it divisible by 10? (a) 4 (b) 5 (c) 7 (d) 6
32. 32.What is the units digit of 9^11? (a) 7 (b) 1 (c) 9 (d) 8
33. 33.What is the units digit of 3^55? (a) 3 (b) 7 (c) 9 (d) 1
34. 34.What is the units digit of 7^31? (a) 9 (b) 1 (c) 7 (d) 3
35. 35.Which of these numbers is divisible by 4? (a) 3235 (b) 3228 (c) 3227 (d) 3230
36. 36.What is the units digit of 2^56? (a) 4 (b) 6 (c) 8 (d) 2
37. 37.Which of these numbers is divisible by 3? (a) 1409 (b) 1410 (c) 1411 (d) 1415
38. 38.What is the smallest number you must add to 69 to make it divisible by 4? (a) 2 (b) 3 (c) 1
39. 39.What is the units digit of 3^56? (a) 3 (b) 9 (c) 7 (d) 1
40. 40.Which of these numbers is divisible by 6? (a) 1581 (b) 1572 (c) 1573 (d) 1568

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