Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 807 · 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 4^45? (a) 4 (b) 5 (c) 3 (d) 6
2. 2.Which of these numbers is divisible by 4? (a) 323 (b) 329 (c) 324 (d) 326
3. 3.Which of these numbers is divisible by 3? (a) 2686 (b) 2693 (c) 2688 (d) 2689
4. 4.Which of these numbers is divisible by 3? (a) 1900 (b) 1897 (c) 1896 (d) 1895
5. 5.What is the smallest number you must add to 147 to make it divisible by 12? (a) 3 (b) 9 (c) 8 (d) 10
6. 6.What is the remainder when 289 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
7. 7.What is the remainder when 466 is divided by 8? (a) 1 (b) 2 (c) 3 (d) 6
8. 8.What is the smallest number you must add to 75 to make it divisible by 6? (a) 5 (b) 2 (c) 3 (d) 4
9. 9.Which of these numbers is divisible by 6? (a) 4563 (b) 4575 (c) 4567 (d) 4566
10. 10.What is the smallest number you must add to 356 to make it divisible by 12? (a) 8 (b) 4 (c) 5 (d) 3
11. 11.What is the remainder when 47 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
12. 12.Which of these numbers is divisible by 9? (a) 1554 (b) 1562 (c) 1548 (d) 1544
13. 13.Which of these numbers is divisible by 11? (a) 8629 (b) 8624 (c) 8623 (d) 8641
14. 14.What is the smallest number you must add to 274 to make it divisible by 9? (a) 7 (b) 5 (c) 4 (d) 6
15. 15.Which of these numbers is divisible by 3? (a) 487 (b) 485 (c) 483 (d) 481
16. 16.What is the smallest number you must add to 391 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
17. 17.What is the units digit of 7^60? (a) 7 (b) 1 (c) 3 (d) 9
18. 18.Which of these numbers is divisible by 3? (a) 2566 (b) 2564 (c) 2569 (d) 2565
19. 19.What is the smallest number you must add to 362 to make it divisible by 5? (a) 4 (b) 1 (c) 2 (d) 3
20. 20.What is the units digit of 3^51? (a) 7 (b) 3 (c) 1 (d) 9
21. 21.Which of these numbers is divisible by 4? (a) 3236 (b) 3235 (c) 3238 (d) 3243
22. 22.What is the remainder when 486 is divided by 7? (a) 5 (b) 4 (c) 3 (d) 2
23. 23.What is the units digit of 3^40? (a) 3 (b) 9 (c) 7 (d) 1
24. 24.What is the remainder when 214 is divided by 3? (a) 0 (b) 2 (c) 1
25. 25.Which of these numbers is divisible by 9? (a) 379 (b) 376 (c) 391 (d) 378
26. 26.Which of these numbers is divisible by 6? (a) 3805 (b) 3817 (c) 3813 (d) 3810
27. 27.What is the units digit of 4^48? (a) 7 (b) 4 (c) 5 (d) 6
28. 28.What is the units digit of 7^12? (a) 7 (b) 1 (c) 3 (d) 9
29. 29.Which of these numbers is divisible by 9? (a) 3793 (b) 3788 (c) 3780 (d) 3772
30. 30.What is the remainder when 606 is divided by 8? (a) 5 (b) 7 (c) 6 (d) 2
31. 31.What is the units digit of 3^16? (a) 9 (b) 1 (c) 7 (d) 3
32. 32.Which of these numbers is divisible by 3? (a) 892 (b) 887 (c) 888 (d) 890
33. 33.What is the remainder when 410 is divided by 8? (a) 6 (b) 3 (c) 1 (d) 2
34. 34.What is the units digit of 3^47? (a) 3 (b) 7 (c) 1 (d) 9
35. 35.Which of these numbers is divisible by 11? (a) 3116 (b) 3124 (c) 3130 (d) 3143
36. 36.What is the remainder when 279 is divided by 4? (a) 1 (b) 3 (c) 2 (d) 0
37. 37.What is the smallest number you must add to 818 to make it divisible by 11? (a) 4 (b) 8 (c) 6 (d) 7
38. 38.What is the remainder when 297 is divided by 8? (a) 0 (b) 2 (c) 1 (d) 7
39. 39.What is the remainder when 514 is divided by 9? (a) 1 (b) 0 (c) 8 (d) 2
40. 40.What is the units digit of 8^53? (a) 6 (b) 8 (c) 4 (d) 2

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