Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 249 · 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 units digit of 4^48? (a) 5 (b) 4 (c) 7 (d) 6
2. 2.What is the units digit of 4^30? (a) 5 (b) 7 (c) 6 (d) 4
3. 3.Which of these numbers is divisible by 3? (a) 707 (b) 701 (c) 702 (d) 704
4. 4.Which of these numbers is divisible by 4? (a) 3319 (b) 3320 (c) 3325 (d) 3321
5. 5.What is the smallest number you must add to 291 to make it divisible by 4? (a) 1 (b) 3 (c) 2
6. 6.What is the remainder when 351 is divided by 8? (a) 7 (b) 6 (c) 5 (d) 1
7. 7.Which of these numbers is divisible by 6? (a) 4808 (b) 4801 (c) 4817 (d) 4806
8. 8.What is the smallest number you must add to 733 to make it divisible by 12? (a) 10 (b) 9 (c) 11 (d) 1
9. 9.What is the units digit of 3^55? (a) 1 (b) 9 (c) 7 (d) 3
10. 10.What is the smallest number you must add to 236 to make it divisible by 10? (a) 3 (b) 5 (c) 4 (d) 6
11. 11.What is the smallest number you must add to 348 to make it divisible by 11? (a) 4 (b) 7 (c) 3 (d) 5
12. 12.Which of these numbers is divisible by 6? (a) 2363 (b) 2373 (c) 2368 (d) 2364
13. 13.What is the remainder when 245 is divided by 6? (a) 3 (b) 1 (c) 4 (d) 5
14. 14.What is the remainder when 460 is divided by 7? (a) 4 (b) 6 (c) 2 (d) 5
15. 15.What is the units digit of 2^45? (a) 6 (b) 8 (c) 4 (d) 2
16. 16.What is the smallest number you must add to 363 to make it divisible by 10? (a) 7 (b) 6 (c) 3 (d) 8
17. 17.What is the smallest number you must add to 137 to make it divisible by 4? (a) 3 (b) 2 (c) 1
18. 18.What is the smallest number you must add to 718 to make it divisible by 12? (a) 10 (b) 1 (c) 3 (d) 2
19. 19.What is the smallest number you must add to 655 to make it divisible by 10? (a) 6 (b) 7 (c) 5 (d) 4
20. 20.What is the remainder when 469 is divided by 6? (a) 2 (b) 5 (c) 0 (d) 1
21. 21.What is the remainder when 568 is divided by 9? (a) 0 (b) 8 (c) 1 (d) 2
22. 22.What is the units digit of 2^57? (a) 8 (b) 4 (c) 2 (d) 6
23. 23.What is the remainder when 153 is divided by 6? (a) 4 (b) 5 (c) 2 (d) 3
24. 24.Which of these numbers is divisible by 6? (a) 2504 (b) 2500 (c) 2510 (d) 2502
25. 25.What is the smallest number you must add to 436 to make it divisible by 7? (a) 6 (b) 4 (c) 5 (d) 2
26. 26.What is the units digit of 7^15? (a) 3 (b) 1 (c) 9 (d) 7
27. 27.What is the remainder when 172 is divided by 10? (a) 2 (b) 1 (c) 3 (d) 8
28. 28.What is the units digit of 4^41? (a) 6 (b) 3 (c) 4 (d) 5
29. 29.What is the units digit of 3^54? (a) 3 (b) 1 (c) 9 (d) 7
30. 30.What is the remainder when 324 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
31. 31.What is the units digit of 9^33? (a) 8 (b) 1 (c) 9 (d) 7
32. 32.Which of these numbers is divisible by 4? (a) 2765 (b) 2760 (c) 2762 (d) 2758
33. 33.What is the smallest number you must add to 587 to make it divisible by 11? (a) 4 (b) 7 (c) 8 (d) 6
34. 34.Which of these numbers is divisible by 11? (a) 7513 (b) 7525 (c) 7515 (d) 7508
35. 35.Which of these numbers is divisible by 9? (a) 1656 (b) 1660 (c) 1651 (d) 1666
36. 36.What is the units digit of 7^31? (a) 7 (b) 3 (c) 1 (d) 9
37. 37.What is the smallest number you must add to 961 to make it divisible by 12? (a) 9 (b) 10 (c) 1 (d) 11
38. 38.What is the units digit of 4^33? (a) 4 (b) 3 (c) 6 (d) 5
39. 39.What is the smallest number you must add to 187 to make it divisible by 6? (a) 3 (b) 4 (c) 5 (d) 1
40. 40.Which of these numbers is divisible by 3? (a) 1610 (b) 1608 (c) 1607 (d) 1613

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