Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 84 · 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 545 to make it divisible by 12? (a) 6 (b) 7 (c) 8 (d) 5
2. 2.What is the units digit of 7^17? (a) 9 (b) 3 (c) 7 (d) 1
3. 3.Which of these numbers is divisible by 3? (a) 2492 (b) 2498 (c) 2493 (d) 2494
4. 4.Which of these numbers is divisible by 3? (a) 2483 (b) 2488 (c) 2484 (d) 2485
5. 5.What is the smallest number you must add to 171 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 2
6. 6.What is the units digit of 3^35? (a) 9 (b) 1 (c) 3 (d) 7
7. 7.What is the remainder when 270 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
8. 8.What is the smallest number you must add to 161 to make it divisible by 3? (a) 2 (b) 1
9. 9.What is the remainder when 694 is divided by 11? (a) 1 (b) 2 (c) 0 (d) 10
10. 10.What is the remainder when 873 is divided by 11? (a) 4 (b) 5 (c) 3 (d) 7
11. 11.What is the remainder when 565 is divided by 12? (a) 0 (b) 11 (c) 1 (d) 2
12. 12.Which of these numbers is divisible by 11? (a) 3124 (b) 3137 (c) 3119 (d) 3126
13. 13.Which of these numbers is divisible by 11? (a) 1177 (b) 1169 (c) 1178 (d) 1198
14. 14.What is the units digit of 7^11? (a) 7 (b) 9 (c) 1 (d) 3
15. 15.What is the remainder when 161 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
16. 16.What is the remainder when 559 is divided by 8? (a) 1 (b) 5 (c) 7 (d) 6
17. 17.What is the smallest number you must add to 223 to make it divisible by 8? (a) 1 (b) 3 (c) 2 (d) 7
18. 18.What is the smallest number you must add to 530 to make it divisible by 12? (a) 9 (b) 2 (c) 11 (d) 10
19. 19.What is the units digit of 8^18? (a) 6 (b) 4 (c) 8 (d) 2
20. 20.What is the remainder when 103 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
21. 21.What is the units digit of 3^37? (a) 1 (b) 7 (c) 3 (d) 9
22. 22.Which of these numbers is divisible by 6? (a) 2533 (b) 2521 (c) 2531 (d) 2526
23. 23.Which of these numbers is divisible by 9? (a) 1972 (b) 1962 (c) 1959 (d) 1969
24. 24.What is the smallest number you must add to 412 to make it divisible by 9? (a) 3 (b) 7 (c) 1 (d) 2
25. 25.Which of these numbers is divisible by 11? (a) 2843 (b) 2836 (c) 2858 (d) 2838
26. 26.Which of these numbers is divisible by 9? (a) 3720 (b) 3728 (c) 3726 (d) 3742
27. 27.What is the smallest number you must add to 327 to make it divisible by 9? (a) 7 (b) 6 (c) 5 (d) 3
28. 28.Which of these numbers is divisible by 11? (a) 3741 (b) 3740 (c) 3737 (d) 3754
29. 29.Which of these numbers is divisible by 6? (a) 3387 (b) 3384 (c) 3391 (d) 3380
30. 30.What is the remainder when 248 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
31. 31.What is the remainder when 377 is divided by 7? (a) 6 (b) 5 (c) 4 (d) 1
32. 32.What is the smallest number you must add to 40 to make it divisible by 3? (a) 1 (b) 2
33. 33.What is the smallest number you must add to 80 to make it divisible by 3? (a) 1 (b) 2
34. 34.Which of these numbers is divisible by 8? (a) 4315 (b) 4302 (c) 4304 (d) 4310
35. 35.What is the units digit of 7^56? (a) 9 (b) 1 (c) 7 (d) 3
36. 36.Which of these numbers is divisible by 3? (a) 881 (b) 884 (c) 879 (d) 878
37. 37.What is the units digit of 3^10? (a) 9 (b) 7 (c) 3 (d) 1
38. 38.What is the remainder when 692 is divided by 10? (a) 2 (b) 3 (c) 1 (d) 8
39. 39.Which of these numbers is divisible by 6? (a) 2335 (b) 2340 (c) 2341 (d) 2349
40. 40.Which of these numbers is divisible by 4? (a) 1692 (b) 1697 (c) 1695 (d) 1690

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