Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 504 · 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 remainder when 434 is divided by 6? (a) 4 (b) 1 (c) 2 (d) 3
2. 2.Which of these numbers is divisible by 11? (a) 1742 (b) 1733 (c) 1756 (d) 1738
3. 3.What is the units digit of 9^49? (a) 8 (b) 7 (c) 9 (d) 1
4. 4.Which of these numbers is divisible by 4? (a) 2261 (b) 2265 (c) 2269 (d) 2264
5. 5.What is the smallest number you must add to 341 to make it divisible by 9? (a) 8 (b) 2 (c) 1 (d) 3
6. 6.What is the smallest number you must add to 110 to make it divisible by 4? (a) 2 (b) 3 (c) 1
7. 7.Which of these numbers is divisible by 3? (a) 411 (b) 409 (c) 415 (d) 412
8. 8.What is the units digit of 2^49? (a) 8 (b) 2 (c) 4 (d) 6
9. 9.What is the smallest number you must add to 762 to make it divisible by 11? (a) 7 (b) 9 (c) 8 (d) 3
10. 10.Which of these numbers is divisible by 11? (a) 4760 (b) 4780 (c) 4769 (d) 4763
11. 11.What is the smallest number you must add to 37 to make it divisible by 3? (a) 1 (b) 2
12. 12.Which of these numbers is divisible by 11? (a) 3872 (b) 3866 (c) 3890 (d) 3874
13. 13.What is the smallest number you must add to 677 to make it divisible by 12? (a) 5 (b) 6 (c) 7 (d) 8
14. 14.What is the units digit of 2^16? (a) 8 (b) 2 (c) 6 (d) 4
15. 15.Which of these numbers is divisible by 8? (a) 6203 (b) 6215 (c) 6194 (d) 6200
16. 16.Which of these numbers is divisible by 11? (a) 1535 (b) 1518 (c) 1516 (d) 1520
17. 17.Which of these numbers is divisible by 8? (a) 3815 (b) 3827 (c) 3816 (d) 3823
18. 18.Which of these numbers is divisible by 3? (a) 886 (b) 893 (c) 888 (d) 889
19. 19.What is the smallest number you must add to 823 to make it divisible by 11? (a) 2 (b) 1 (c) 3 (d) 9
20. 20.What is the smallest number you must add to 64 to make it divisible by 3? (a) 1 (b) 2
21. 21.What is the remainder when 552 is divided by 11? (a) 3 (b) 2 (c) 9 (d) 1
22. 22.What is the units digit of 2^41? (a) 2 (b) 8 (c) 4 (d) 6
23. 23.What is the smallest number you must add to 73 to make it divisible by 7? (a) 3 (b) 5 (c) 6 (d) 4
24. 24.Which of these numbers is divisible by 6? (a) 4024 (b) 4031 (c) 4017 (d) 4020
25. 25.Which of these numbers is divisible by 8? (a) 3555 (b) 3546 (c) 3544 (d) 3542
26. 26.What is the remainder when 87 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
27. 27.Which of these numbers is divisible by 3? (a) 2305 (b) 2300 (c) 2301 (d) 2302
28. 28.What is the remainder when 93 is divided by 9? (a) 4 (b) 2 (c) 3 (d) 6
29. 29.What is the remainder when 83 is divided by 6? (a) 1 (b) 5 (c) 3 (d) 4
30. 30.What is the smallest number you must add to 283 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
31. 31.What is the units digit of 3^54? (a) 3 (b) 7 (c) 1 (d) 9
32. 32.What is the units digit of 9^31? (a) 1 (b) 7 (c) 8 (d) 9
33. 33.What is the remainder when 316 is divided by 12? (a) 4 (b) 5 (c) 3 (d) 8
34. 34.Which of these numbers is divisible by 8? (a) 5283 (b) 5271 (c) 5277 (d) 5272
35. 35.Which of these numbers is divisible by 4? (a) 2528 (b) 2527 (c) 2530 (d) 2534
36. 36.What is the units digit of 4^19? (a) 3 (b) 4 (c) 5 (d) 6
37. 37.What is the units digit of 8^17? (a) 4 (b) 6 (c) 2 (d) 8
38. 38.What is the units digit of 3^50? (a) 3 (b) 1 (c) 7 (d) 9
39. 39.What is the units digit of 7^30? (a) 3 (b) 1 (c) 9 (d) 7
40. 40.Which of these numbers is divisible by 11? (a) 3489 (b) 3477 (c) 3506 (d) 3487

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