Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 587 · 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 4^20? (a) 4 (b) 7 (c) 5 (d) 6
2. 2.What is the smallest number you must add to 527 to make it divisible by 10? (a) 7 (b) 2 (c) 3 (d) 4
3. 3.What is the remainder when 278 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
4. 4.What is the smallest number you must add to 565 to make it divisible by 11? (a) 8 (b) 7 (c) 6 (d) 4
5. 5.What is the remainder when 169 is divided by 8? (a) 2 (b) 1 (c) 0 (d) 7
6. 6.What is the remainder when 502 is divided by 8? (a) 7 (b) 5 (c) 6 (d) 2
7. 7.What is the units digit of 3^41? (a) 1 (b) 7 (c) 3 (d) 9
8. 8.Which of these numbers is divisible by 3? (a) 2578 (b) 2577 (c) 2581 (d) 2575
9. 9.Which of these numbers is divisible by 8? (a) 4649 (b) 4656 (c) 4667 (d) 4661
10. 10.Which of these numbers is divisible by 3? (a) 1512 (b) 1510 (c) 1513 (d) 1517
11. 11.What is the units digit of 7^20? (a) 1 (b) 9 (c) 7 (d) 3
12. 12.What is the smallest number you must add to 131 to make it divisible by 9? (a) 4 (b) 6 (c) 3 (d) 5
13. 13.What is the smallest number you must add to 61 to make it divisible by 3? (a) 1 (b) 2
14. 14.What is the remainder when 232 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
15. 15.What is the smallest number you must add to 130 to make it divisible by 3? (a) 2 (b) 1
16. 16.What is the units digit of 9^51? (a) 7 (b) 8 (c) 9 (d) 1
17. 17.What is the units digit of 7^14? (a) 3 (b) 1 (c) 7 (d) 9
18. 18.What is the remainder when 332 is divided by 11? (a) 3 (b) 9 (c) 1 (d) 2
19. 19.What is the smallest number you must add to 667 to make it divisible by 10? (a) 7 (b) 4 (c) 2 (d) 3
20. 20.What is the smallest number you must add to 861 to make it divisible by 11? (a) 3 (b) 8 (c) 9 (d) 7
21. 21.Which of these numbers is divisible by 6? (a) 1740 (b) 1736 (c) 1750 (d) 1744
22. 22.What is the smallest number you must add to 353 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
23. 23.What is the remainder when 259 is divided by 8? (a) 2 (b) 3 (c) 4 (d) 5
24. 24.What is the units digit of 3^49? (a) 3 (b) 7 (c) 1 (d) 9
25. 25.What is the remainder when 183 is divided by 7? (a) 0 (b) 1 (c) 2 (d) 6
26. 26.What is the units digit of 9^10? (a) 9 (b) 2 (c) 0 (d) 1
27. 27.What is the units digit of 2^18? (a) 2 (b) 8 (c) 4 (d) 6
28. 28.What is the remainder when 178 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
29. 29.What is the remainder when 461 is divided by 9? (a) 7 (b) 2 (c) 1 (d) 3
30. 30.What is the units digit of 4^31? (a) 3 (b) 6 (c) 4 (d) 5
31. 31.What is the remainder when 359 is divided by 6? (a) 1 (b) 4 (c) 5 (d) 3
32. 32.Which of these numbers is divisible by 8? (a) 3114 (b) 3107 (c) 3101 (d) 3104
33. 33.What is the remainder when 349 is divided by 7? (a) 5 (b) 6 (c) 4 (d) 1
34. 34.Which of these numbers is divisible by 4? (a) 1280 (b) 1278 (c) 1283 (d) 1287
35. 35.Which of these numbers is divisible by 6? (a) 3197 (b) 3189 (c) 3185 (d) 3186
36. 36.Which of these numbers is divisible by 4? (a) 3376 (b) 3383 (c) 3375 (d) 3377
37. 37.Which of these numbers is divisible by 3? (a) 1429 (b) 1431 (c) 1435 (d) 1432
38. 38.What is the units digit of 9^31? (a) 8 (b) 7 (c) 1 (d) 9
39. 39.What is the remainder when 214 is divided by 3? (a) 2 (b) 0 (c) 1
40. 40.Which of these numbers is divisible by 9? (a) 904 (b) 891 (c) 898 (d) 887

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