Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 912 · 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 236 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
2. 2.Which of these numbers is divisible by 4? (a) 751 (b) 754 (c) 758 (d) 752
3. 3.What is the units digit of 3^26? (a) 7 (b) 1 (c) 9 (d) 3
4. 4.Which of these numbers is divisible by 9? (a) 5139 (b) 5132 (c) 5149 (d) 5147
5. 5.Which of these numbers is divisible by 6? (a) 3959 (b) 3948 (c) 3946 (d) 3951
6. 6.What is the remainder when 76 is divided by 3? (a) 0 (b) 1 (c) 2
7. 7.Which of these numbers is divisible by 6? (a) 2499 (b) 2485 (c) 2490 (d) 2492
8. 8.What is the smallest number you must add to 540 to make it divisible by 8? (a) 3 (b) 6 (c) 5 (d) 4
9. 9.What is the units digit of 3^49? (a) 9 (b) 3 (c) 7 (d) 1
10. 10.What is the units digit of 2^20? (a) 6 (b) 2 (c) 4 (d) 8
11. 11.What is the smallest number you must add to 285 to make it divisible by 4? (a) 1 (b) 3 (c) 2
12. 12.What is the smallest number you must add to 242 to make it divisible by 8? (a) 6 (b) 5 (c) 2 (d) 7
13. 13.Which of these numbers is divisible by 6? (a) 5286 (b) 5296 (c) 5284 (d) 5288
14. 14.Which of these numbers is divisible by 6? (a) 373 (b) 365 (c) 368 (d) 366
15. 15.What is the remainder when 152 is divided by 3? (a) 0 (b) 2 (c) 1
16. 16.What is the units digit of 3^40? (a) 1 (b) 3 (c) 7 (d) 9
17. 17.Which of these numbers is divisible by 9? (a) 3071 (b) 3078 (c) 3088 (d) 3079
18. 18.What is the smallest number you must add to 618 to make it divisible by 11? (a) 8 (b) 9 (c) 2 (d) 10
19. 19.Which of these numbers is divisible by 8? (a) 865 (b) 877 (c) 860 (d) 864
20. 20.What is the smallest number you must add to 146 to make it divisible by 7? (a) 1 (b) 6 (c) 2 (d) 3
21. 21.What is the remainder when 151 is divided by 11? (a) 3 (b) 7 (c) 9 (d) 8
22. 22.Which of these numbers is divisible by 6? (a) 3584 (b) 3599 (c) 3588 (d) 3592
23. 23.What is the units digit of 4^26? (a) 5 (b) 7 (c) 4 (d) 6
24. 24.What is the smallest number you must add to 74 to make it divisible by 3? (a) 2 (b) 1
25. 25.What is the smallest number you must add to 519 to make it divisible by 7? (a) 5 (b) 1 (c) 4 (d) 6
26. 26.What is the smallest number you must add to 204 to make it divisible by 10? (a) 6 (b) 5 (c) 7 (d) 4
27. 27.What is the smallest number you must add to 553 to make it divisible by 9? (a) 5 (b) 7 (c) 4 (d) 6
28. 28.What is the remainder when 263 is divided by 12? (a) 9 (b) 1 (c) 11 (d) 10
29. 29.What is the units digit of 4^14? (a) 4 (b) 7 (c) 6 (d) 5
30. 30.What is the remainder when 241 is divided by 10? (a) 2 (b) 9 (c) 1 (d) 0
31. 31.What is the units digit of 7^38? (a) 9 (b) 3 (c) 7 (d) 1
32. 32.What is the remainder when 165 is divided by 8? (a) 3 (b) 6 (c) 5 (d) 4
33. 33.Which of these numbers is divisible by 3? (a) 432 (b) 430 (c) 433 (d) 436
34. 34.Which of these numbers is divisible by 6? (a) 4091 (b) 4093 (c) 4083 (d) 4086
35. 35.What is the remainder when 498 is divided by 11? (a) 4 (b) 3 (c) 2 (d) 8
36. 36.What is the remainder when 101 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 0
37. 37.What is the remainder when 300 is divided by 8? (a) 5 (b) 3 (c) 6 (d) 4
38. 38.What is the units digit of 7^28? (a) 3 (b) 9 (c) 7 (d) 1
39. 39.What is the remainder when 314 is divided by 6? (a) 4 (b) 1 (c) 2 (d) 3
40. 40.What is the units digit of 4^24? (a) 6 (b) 4 (c) 5 (d) 7

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