Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 825 · 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 9? (a) 2703 (b) 2719 (c) 2709 (d) 2713
2. 2.Which of these numbers is divisible by 9? (a) 5720 (b) 5715 (c) 5713 (d) 5727
3. 3.Which of these numbers is divisible by 3? (a) 219 (b) 221 (c) 223 (d) 217
4. 4.Which of these numbers is divisible by 3? (a) 2119 (b) 2117 (c) 2115 (d) 2113
5. 5.What is the units digit of 4^46? (a) 5 (b) 6 (c) 4 (d) 7
6. 6.What is the remainder when 369 is divided by 11? (a) 5 (b) 6 (c) 8 (d) 7
7. 7.What is the remainder when 172 is divided by 3? (a) 1 (b) 0 (c) 2
8. 8.What is the units digit of 8^51? (a) 8 (b) 4 (c) 6 (d) 2
9. 9.Which of these numbers is divisible by 6? (a) 454 (b) 464 (c) 461 (d) 456
10. 10.What is the smallest number you must add to 656 to make it divisible by 9? (a) 8 (b) 2 (c) 1 (d) 3
11. 11.What is the units digit of 9^24? (a) 2 (b) 0 (c) 9 (d) 1
12. 12.Which of these numbers is divisible by 3? (a) 1912 (b) 1909 (c) 1908 (d) 1907
13. 13.What is the smallest number you must add to 350 to make it divisible by 12? (a) 11 (b) 9 (c) 10 (d) 2
14. 14.What is the units digit of 8^38? (a) 2 (b) 4 (c) 8 (d) 6
15. 15.What is the units digit of 9^59? (a) 7 (b) 8 (c) 9 (d) 1
16. 16.Which of these numbers is divisible by 3? (a) 703 (b) 698 (c) 699 (d) 701
17. 17.Which of these numbers is divisible by 9? (a) 3329 (b) 3305 (c) 3315 (d) 3312
18. 18.What is the smallest number you must add to 601 to make it divisible by 10? (a) 1 (b) 7 (c) 9 (d) 8
19. 19.What is the smallest number you must add to 759 to make it divisible by 10? (a) 9 (b) 1 (c) 3 (d) 2
20. 20.What is the remainder when 196 is divided by 3? (a) 0 (b) 2 (c) 1
21. 21.What is the smallest number you must add to 279 to make it divisible by 4? (a) 3 (b) 2 (c) 1
22. 22.Which of these numbers is divisible by 8? (a) 4544 (b) 4548 (c) 4557 (d) 4537
23. 23.What is the smallest number you must add to 637 to make it divisible by 9? (a) 2 (b) 3 (c) 7 (d) 1
24. 24.Which of these numbers is divisible by 11? (a) 3750 (b) 3731 (c) 3729 (d) 3724
25. 25.What is the remainder when 605 is divided by 10? (a) 7 (b) 4 (c) 6 (d) 5
26. 26.What is the units digit of 9^21? (a) 8 (b) 7 (c) 9 (d) 1
27. 27.What is the units digit of 4^19? (a) 6 (b) 5 (c) 3 (d) 4
28. 28.What is the units digit of 3^16? (a) 7 (b) 3 (c) 9 (d) 1
29. 29.Which of these numbers is divisible by 6? (a) 5002 (b) 4987 (c) 4994 (d) 4992
30. 30.What is the units digit of 3^30? (a) 3 (b) 7 (c) 9 (d) 1
31. 31.What is the remainder when 314 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
32. 32.What is the units digit of 2^54? (a) 2 (b) 6 (c) 8 (d) 4
33. 33.What is the remainder when 562 is divided by 8? (a) 3 (b) 2 (c) 1 (d) 6
34. 34.What is the smallest number you must add to 249 to make it divisible by 4? (a) 2 (b) 3 (c) 1
35. 35.What is the units digit of 9^60? (a) 9 (b) 2 (c) 1 (d) 0
36. 36.What is the units digit of 8^22? (a) 8 (b) 2 (c) 4 (d) 6
37. 37.What is the remainder when 117 is divided by 7? (a) 2 (b) 6 (c) 5 (d) 4
38. 38.Which of these numbers is divisible by 8? (a) 3288 (b) 3290 (c) 3282 (d) 3303
39. 39.What is the smallest number you must add to 236 to make it divisible by 10? (a) 4 (b) 3 (c) 5 (d) 6
40. 40.Which of these numbers is divisible by 11? (a) 1226 (b) 1251 (c) 1241 (d) 1232

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