Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 230 · 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) 3765 (b) 3757 (c) 3753 (d) 3756
2. 2.Which of these numbers is divisible by 4? (a) 3183 (b) 3177 (c) 3176 (d) 3174
3. 3.Which of these numbers is divisible by 3? (a) 1103 (b) 1105 (c) 1101 (d) 1100
4. 4.What is the units digit of 2^28? (a) 2 (b) 8 (c) 6 (d) 4
5. 5.What is the remainder when 107 is divided by 3? (a) 1 (b) 2 (c) 0
6. 6.What is the remainder when 265 is divided by 8? (a) 7 (b) 2 (c) 1 (d) 0
7. 7.What is the remainder when 487 is divided by 11? (a) 3 (b) 4 (c) 2 (d) 8
8. 8.Which of these numbers is divisible by 4? (a) 2644 (b) 2643 (c) 2646 (d) 2650
9. 9.What is the units digit of 4^11? (a) 4 (b) 6 (c) 5 (d) 3
10. 10.What is the remainder when 425 is divided by 9? (a) 7 (b) 2 (c) 3 (d) 1
11. 11.Which of these numbers is divisible by 4? (a) 654 (b) 652 (c) 649 (d) 659
12. 12.What is the units digit of 4^20? (a) 5 (b) 6 (c) 7 (d) 4
13. 13.What is the smallest number you must add to 169 to make it divisible by 3? (a) 1 (b) 2
14. 14.Which of these numbers is divisible by 8? (a) 5238 (b) 5231 (c) 5221 (d) 5224
15. 15.What is the units digit of 9^50? (a) 1 (b) 9 (c) 0 (d) 2
16. 16.Which of these numbers is divisible by 9? (a) 5443 (b) 5451 (c) 5461 (d) 5445
17. 17.What is the remainder when 197 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
18. 18.What is the smallest number you must add to 295 to make it divisible by 12? (a) 4 (b) 7 (c) 6 (d) 5
19. 19.What is the remainder when 626 is divided by 10? (a) 6 (b) 4 (c) 5 (d) 7
20. 20.What is the smallest number you must add to 314 to make it divisible by 4? (a) 1 (b) 2 (c) 3
21. 21.What is the smallest number you must add to 371 to make it divisible by 10? (a) 1 (b) 8 (c) 7 (d) 9
22. 22.What is the remainder when 254 is divided by 10? (a) 5 (b) 3 (c) 4 (d) 6
23. 23.Which of these numbers is divisible by 4? (a) 3416 (b) 3414 (c) 3419 (d) 3423
24. 24.Which of these numbers is divisible by 8? (a) 3604 (b) 3623 (c) 3608 (d) 3613
25. 25.What is the units digit of 2^42? (a) 6 (b) 8 (c) 4 (d) 2
26. 26.Which of these numbers is divisible by 4? (a) 2774 (b) 2778 (c) 2782 (d) 2776
27. 27.What is the remainder when 870 is divided by 12? (a) 7 (b) 5 (c) 8 (d) 6
28. 28.What is the remainder when 220 is divided by 7? (a) 5 (b) 4 (c) 3 (d) 2
29. 29.Which of these numbers is divisible by 9? (a) 1802 (b) 1798 (c) 1814 (d) 1800
30. 30.What is the smallest number you must add to 142 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
31. 31.What is the smallest number you must add to 446 to make it divisible by 8? (a) 1 (b) 3 (c) 6 (d) 2
32. 32.What is the smallest number you must add to 235 to make it divisible by 3? (a) 1 (b) 2
33. 33.What is the units digit of 3^46? (a) 9 (b) 3 (c) 1 (d) 7
34. 34.What is the units digit of 8^21? (a) 6 (b) 4 (c) 2 (d) 8
35. 35.What is the units digit of 3^51? (a) 9 (b) 3 (c) 7 (d) 1
36. 36.What is the units digit of 3^37? (a) 9 (b) 1 (c) 3 (d) 7
37. 37.What is the remainder when 238 is divided by 12? (a) 9 (b) 10 (c) 11 (d) 2
38. 38.What is the smallest number you must add to 505 to make it divisible by 8? (a) 1 (b) 5 (c) 6 (d) 7
39. 39.What is the smallest number you must add to 297 to make it divisible by 6? (a) 3 (b) 5 (c) 4 (d) 2
40. 40.What is the remainder when 179 is divided by 3? (a) 1 (b) 0 (c) 2

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