Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 117 · 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 107 to make it divisible by 3? (a) 1 (b) 2
2. 2.Which of these numbers is divisible by 9? (a) 3421 (b) 3430 (c) 3420 (d) 3419
3. 3.Which of these numbers is divisible by 4? (a) 756 (b) 761 (c) 759 (d) 753
4. 4.Which of these numbers is divisible by 4? (a) 173 (b) 181 (c) 176 (d) 179
5. 5.Which of these numbers is divisible by 6? (a) 3130 (b) 3142 (c) 3132 (d) 3137
6. 6.What is the remainder when 781 is divided by 10? (a) 2 (b) 0 (c) 1 (d) 9
7. 7.What is the units digit of 7^19? (a) 7 (b) 9 (c) 3 (d) 1
8. 8.What is the remainder when 540 is divided by 8? (a) 4 (b) 3 (c) 6 (d) 5
9. 9.Which of these numbers is divisible by 9? (a) 6457 (b) 6444 (c) 6446 (d) 6443
10. 10.Which of these numbers is divisible by 11? (a) 6662 (b) 6645 (c) 6644 (d) 6634
11. 11.What is the units digit of 2^35? (a) 2 (b) 4 (c) 6 (d) 8
12. 12.What is the remainder when 747 is divided by 12? (a) 9 (b) 4 (c) 3 (d) 2
13. 13.What is the smallest number you must add to 64 to make it divisible by 3? (a) 2 (b) 1
14. 14.What is the smallest number you must add to 305 to make it divisible by 7? (a) 5 (b) 3 (c) 2 (d) 4
15. 15.What is the remainder when 586 is divided by 10? (a) 5 (b) 4 (c) 7 (d) 6
16. 16.What is the smallest number you must add to 622 to make it divisible by 9? (a) 6 (b) 7 (c) 1 (d) 8
17. 17.Which of these numbers is divisible by 3? (a) 1694 (b) 1695 (c) 1696 (d) 1699
18. 18.What is the smallest number you must add to 246 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
19. 19.What is the units digit of 2^23? (a) 4 (b) 6 (c) 2 (d) 8
20. 20.What is the units digit of 3^16? (a) 3 (b) 7 (c) 1 (d) 9
21. 21.What is the remainder when 65 is divided by 3? (a) 1 (b) 2 (c) 0
22. 22.What is the smallest number you must add to 345 to make it divisible by 7? (a) 2 (b) 5 (c) 6 (d) 4
23. 23.What is the remainder when 71 is divided by 5? (a) 1 (b) 4 (c) 0 (d) 2
24. 24.Which of these numbers is divisible by 9? (a) 1443 (b) 1439 (c) 1423 (d) 1431
25. 25.What is the units digit of 4^21? (a) 3 (b) 5 (c) 4 (d) 6
26. 26.What is the smallest number you must add to 164 to make it divisible by 8? (a) 3 (b) 6 (c) 4 (d) 5
27. 27.What is the units digit of 9^16? (a) 9 (b) 2 (c) 0 (d) 1
28. 28.What is the remainder when 307 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
29. 29.What is the remainder when 441 is divided by 6? (a) 5 (b) 4 (c) 3 (d) 2
30. 30.What is the units digit of 9^47? (a) 9 (b) 1 (c) 8 (d) 7
31. 31.What is the remainder when 243 is divided by 10? (a) 3 (b) 7 (c) 4 (d) 2
32. 32.What is the units digit of 3^60? (a) 9 (b) 7 (c) 3 (d) 1
33. 33.What is the units digit of 2^54? (a) 8 (b) 4 (c) 6 (d) 2
34. 34.Which of these numbers is divisible by 8? (a) 856 (b) 850 (c) 860 (d) 865
35. 35.What is the smallest number you must add to 728 to make it divisible by 10? (a) 1 (b) 8 (c) 2 (d) 3
36. 36.What is the remainder when 235 is divided by 10? (a) 7 (b) 4 (c) 5 (d) 6
37. 37.What is the remainder when 289 is divided by 12? (a) 2 (b) 0 (c) 11 (d) 1
38. 38.Which of these numbers is divisible by 9? (a) 7402 (b) 7384 (c) 7391 (d) 7389
39. 39.What is the remainder when 304 is divided by 5? (a) 1 (b) 2 (c) 4 (d) 3
40. 40.What is the remainder when 337 is divided by 11? (a) 6 (b) 8 (c) 7 (d) 4

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