Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 82 · 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 4^14? (a) 7 (b) 5 (c) 4 (d) 6
2. 2.Which of these numbers is divisible by 11? (a) 9886 (b) 9892 (c) 9878 (d) 9870
3. 3.What is the units digit of 4^51? (a) 4 (b) 5 (c) 3 (d) 6
4. 4.What is the remainder when 225 is divided by 7? (a) 1 (b) 6 (c) 0 (d) 2
5. 5.What is the smallest number you must add to 403 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
6. 6.What is the smallest number you must add to 290 to make it divisible by 4? (a) 2 (b) 1 (c) 3
7. 7.Which of these numbers is divisible by 4? (a) 390 (b) 395 (c) 386 (d) 388
8. 8.What is the smallest number you must add to 273 to make it divisible by 10? (a) 6 (b) 8 (c) 7 (d) 3
9. 9.What is the units digit of 9^11? (a) 8 (b) 9 (c) 7 (d) 1
10. 10.What is the remainder when 551 is divided by 11? (a) 10 (b) 1 (c) 0 (d) 2
11. 11.What is the remainder when 197 is divided by 7? (a) 0 (b) 1 (c) 6 (d) 2
12. 12.What is the units digit of 3^15? (a) 9 (b) 1 (c) 3 (d) 7
13. 13.What is the units digit of 9^34? (a) 9 (b) 0 (c) 2 (d) 1
14. 14.Which of these numbers is divisible by 11? (a) 1335 (b) 1321 (c) 1320 (d) 1312
15. 15.Which of these numbers is divisible by 8? (a) 5704 (b) 5707 (c) 5714 (d) 5699
16. 16.Which of these numbers is divisible by 11? (a) 3226 (b) 3223 (c) 3238 (d) 3218
17. 17.What is the smallest number you must add to 496 to make it divisible by 7? (a) 6 (b) 1 (c) 2 (d) 3
18. 18.What is the smallest number you must add to 218 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
19. 19.What is the smallest number you must add to 261 to make it divisible by 8? (a) 2 (b) 4 (c) 5 (d) 3
20. 20.Which of these numbers is divisible by 8? (a) 3568 (b) 3571 (c) 3567 (d) 3578
21. 21.What is the remainder when 181 is divided by 11? (a) 4 (b) 7 (c) 6 (d) 5
22. 22.What is the smallest number you must add to 518 to make it divisible by 8? (a) 1 (b) 6 (c) 2 (d) 3
23. 23.What is the smallest number you must add to 174 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
24. 24.What is the smallest number you must add to 91 to make it divisible by 3? (a) 1 (b) 2
25. 25.Which of these numbers is divisible by 4? (a) 1066 (b) 1059 (c) 1060 (d) 1063
26. 26.What is the smallest number you must add to 384 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
27. 27.What is the units digit of 4^28? (a) 7 (b) 4 (c) 5 (d) 6
28. 28.What is the smallest number you must add to 465 to make it divisible by 6? (a) 5 (b) 4 (c) 3 (d) 2
29. 29.What is the remainder when 86 is divided by 7? (a) 5 (b) 1 (c) 2 (d) 3
30. 30.What is the units digit of 4^21? (a) 3 (b) 5 (c) 6 (d) 4
31. 31.What is the remainder when 205 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
32. 32.What is the remainder when 199 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
33. 33.Which of these numbers is divisible by 8? (a) 3291 (b) 3308 (c) 3302 (d) 3296
34. 34.What is the units digit of 3^27? (a) 1 (b) 7 (c) 9 (d) 3
35. 35.Which of these numbers is divisible by 3? (a) 757 (b) 759 (c) 764 (d) 760
36. 36.What is the remainder when 271 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 0
37. 37.What is the remainder when 309 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
38. 38.Which of these numbers is divisible by 8? (a) 1690 (b) 1696 (c) 1702 (d) 1709
39. 39.What is the smallest number you must add to 221 to make it divisible by 12? (a) 7 (b) 8 (c) 5 (d) 6
40. 40.What is the units digit of 8^35? (a) 8 (b) 4 (c) 2 (d) 6

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