Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 541 · 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 units digit of 9^51? (a) 7 (b) 8 (c) 9 (d) 1
2. 2.Which of these numbers is divisible by 3? (a) 332 (b) 338 (c) 333 (d) 334
3. 3.What is the units digit of 3^25? (a) 1 (b) 7 (c) 9 (d) 3
4. 4.Which of these numbers is divisible by 9? (a) 2256 (b) 2247 (c) 2238 (d) 2241
5. 5.Which of these numbers is divisible by 8? (a) 1080 (b) 1076 (c) 1086 (d) 1089
6. 6.What is the units digit of 3^22? (a) 3 (b) 9 (c) 7 (d) 1
7. 7.What is the units digit of 3^28? (a) 9 (b) 7 (c) 1 (d) 3
8. 8.Which of these numbers is divisible by 8? (a) 1192 (b) 1202 (c) 1193 (d) 1186
9. 9.Which of these numbers is divisible by 3? (a) 252 (b) 256 (c) 254 (d) 250
10. 10.What is the units digit of 3^10? (a) 7 (b) 3 (c) 9 (d) 1
11. 11.What is the units digit of 2^39? (a) 4 (b) 2 (c) 6 (d) 8
12. 12.What is the units digit of 3^46? (a) 9 (b) 3 (c) 7 (d) 1
13. 13.What is the smallest number you must add to 352 to make it divisible by 9? (a) 7 (b) 1 (c) 8 (d) 6
14. 14.What is the smallest number you must add to 428 to make it divisible by 11? (a) 2 (b) 1 (c) 10 (d) 3
15. 15.Which of these numbers is divisible by 8? (a) 5054 (b) 5056 (c) 5058 (d) 5067
16. 16.What is the smallest number you must add to 388 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
17. 17.Which of these numbers is divisible by 6? (a) 3017 (b) 3006 (c) 3003 (d) 3008
18. 18.What is the remainder when 263 is divided by 8? (a) 7 (b) 5 (c) 1 (d) 6
19. 19.What is the remainder when 724 is divided by 11? (a) 9 (b) 8 (c) 10 (d) 2
20. 20.What is the remainder when 797 is divided by 10? (a) 3 (b) 7 (c) 8 (d) 6
21. 21.What is the units digit of 4^47? (a) 5 (b) 4 (c) 3 (d) 6
22. 22.Which of these numbers is divisible by 3? (a) 1663 (b) 1666 (c) 1660 (d) 1662
23. 23.What is the remainder when 117 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
24. 24.Which of these numbers is divisible by 3? (a) 626 (b) 631 (c) 627 (d) 629
25. 25.What is the smallest number you must add to 322 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 2
26. 26.What is the units digit of 3^54? (a) 7 (b) 3 (c) 9 (d) 1
27. 27.What is the remainder when 96 is divided by 9? (a) 3 (b) 5 (c) 7 (d) 6
28. 28.What is the smallest number you must add to 254 to make it divisible by 7? (a) 4 (b) 2 (c) 6 (d) 5
29. 29.What is the units digit of 7^19? (a) 9 (b) 1 (c) 7 (d) 3
30. 30.What is the remainder when 133 is divided by 11? (a) 0 (b) 2 (c) 10 (d) 1
31. 31.Which of these numbers is divisible by 11? (a) 9323 (b) 9336 (c) 9340 (d) 9328
32. 32.What is the smallest number you must add to 197 to make it divisible by 10? (a) 3 (b) 4 (c) 7 (d) 2
33. 33.Which of these numbers is divisible by 3? (a) 478 (b) 477 (c) 481 (d) 476
34. 34.Which of these numbers is divisible by 3? (a) 1764 (b) 1768 (c) 1762 (d) 1766
35. 35.What is the units digit of 9^55? (a) 8 (b) 9 (c) 1 (d) 7
36. 36.What is the smallest number you must add to 315 to make it divisible by 4? (a) 2 (b) 3 (c) 1
37. 37.What is the remainder when 234 is divided by 8? (a) 2 (b) 1 (c) 3 (d) 6
38. 38.Which of these numbers is divisible by 11? (a) 1063 (b) 1055 (c) 1056 (d) 1069
39. 39.What is the smallest number you must add to 70 to make it divisible by 4? (a) 1 (b) 2 (c) 3
40. 40.What is the units digit of 7^38? (a) 9 (b) 1 (c) 7 (d) 3

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