Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 596 · 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 3? (a) 169 (b) 167 (c) 173 (d) 168
2. 2.What is the smallest number you must add to 454 to make it divisible by 10? (a) 7 (b) 5 (c) 4 (d) 6
3. 3.What is the units digit of 4^25? (a) 4 (b) 3 (c) 6 (d) 5
4. 4.What is the smallest number you must add to 364 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
5. 5.Which of these numbers is divisible by 4? (a) 826 (b) 822 (c) 824 (d) 831
6. 6.What is the smallest number you must add to 128 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
7. 7.Which of these numbers is divisible by 6? (a) 660 (b) 670 (c) 658 (d) 661
8. 8.What is the units digit of 3^59? (a) 9 (b) 7 (c) 3 (d) 1
9. 9.What is the smallest number you must add to 305 to make it divisible by 4? (a) 2 (b) 3 (c) 1
10. 10.What is the smallest number you must add to 196 to make it divisible by 8? (a) 3 (b) 5 (c) 4 (d) 6
11. 11.Which of these numbers is divisible by 11? (a) 8443 (b) 8457 (c) 8437 (d) 8432
12. 12.What is the smallest number you must add to 226 to make it divisible by 3? (a) 2 (b) 1
13. 13.What is the remainder when 543 is divided by 11? (a) 4 (b) 7 (c) 3 (d) 5
14. 14.What is the remainder when 290 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
15. 15.What is the smallest number you must add to 313 to make it divisible by 4? (a) 1 (b) 3 (c) 2
16. 16.Which of these numbers is divisible by 4? (a) 2067 (b) 2071 (c) 2074 (d) 2068
17. 17.What is the units digit of 4^29? (a) 5 (b) 4 (c) 3 (d) 6
18. 18.What is the units digit of 4^60? (a) 7 (b) 5 (c) 6 (d) 4
19. 19.What is the units digit of 9^13? (a) 7 (b) 1 (c) 9 (d) 8
20. 20.What is the smallest number you must add to 67 to make it divisible by 6? (a) 3 (b) 1 (c) 4 (d) 5
21. 21.What is the smallest number you must add to 215 to make it divisible by 3? (a) 1 (b) 2
22. 22.What is the units digit of 4^11? (a) 3 (b) 5 (c) 6 (d) 4
23. 23.What is the units digit of 7^46? (a) 9 (b) 1 (c) 3 (d) 7
24. 24.What is the remainder when 678 is divided by 10? (a) 7 (b) 2 (c) 9 (d) 8
25. 25.Which of these numbers is divisible by 6? (a) 4379 (b) 4368 (c) 4373 (d) 4365
26. 26.What is the units digit of 9^23? (a) 7 (b) 1 (c) 8 (d) 9
27. 27.What is the smallest number you must add to 31 to make it divisible by 3? (a) 2 (b) 1
28. 28.What is the units digit of 3^37? (a) 7 (b) 9 (c) 3 (d) 1
29. 29.What is the smallest number you must add to 622 to make it divisible by 10? (a) 7 (b) 2 (c) 8 (d) 9
30. 30.What is the smallest number you must add to 178 to make it divisible by 8? (a) 2 (b) 6 (c) 5 (d) 7
31. 31.Which of these numbers is divisible by 6? (a) 757 (b) 765 (c) 754 (d) 756
32. 32.What is the units digit of 3^13? (a) 7 (b) 3 (c) 9 (d) 1
33. 33.What is the units digit of 9^19? (a) 9 (b) 7 (c) 1 (d) 8
34. 34.What is the remainder when 341 is divided by 5? (a) 2 (b) 0 (c) 1 (d) 4
35. 35.Which of these numbers is divisible by 11? (a) 3534 (b) 3561 (c) 3542 (d) 3550
36. 36.What is the units digit of 2^29? (a) 4 (b) 8 (c) 6 (d) 2
37. 37.Which of these numbers is divisible by 4? (a) 2493 (b) 2487 (c) 2488 (d) 2490
38. 38.What is the units digit of 3^57? (a) 3 (b) 1 (c) 9 (d) 7
39. 39.Which of these numbers is divisible by 11? (a) 2793 (b) 2774 (c) 2770 (d) 2772
40. 40.Which of these numbers is divisible by 9? (a) 4657 (b) 4648 (c) 4653 (d) 4667

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