Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 351 · 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 smallest number you must add to 163 to make it divisible by 6? (a) 1 (b) 4 (c) 3 (d) 5
2. 2.What is the remainder when 215 is divided by 11? (a) 7 (b) 6 (c) 5 (d) 8
3. 3.Which of these numbers is divisible by 8? (a) 1634 (b) 1632 (c) 1644 (d) 1626
4. 4.What is the remainder when 395 is divided by 10? (a) 6 (b) 4 (c) 7 (d) 5
5. 5.What is the units digit of 4^60? (a) 6 (b) 4 (c) 7 (d) 5
6. 6.What is the remainder when 146 is divided by 5? (a) 1 (b) 2 (c) 0 (d) 4
7. 7.What is the remainder when 73 is divided by 4? (a) 2 (b) 0 (c) 1 (d) 3
8. 8.What is the smallest number you must add to 191 to make it divisible by 7? (a) 2 (b) 6 (c) 4 (d) 5
9. 9.What is the remainder when 108 is divided by 7? (a) 4 (b) 5 (c) 2 (d) 3
10. 10.What is the units digit of 9^57? (a) 1 (b) 7 (c) 9 (d) 8
11. 11.What is the units digit of 3^18? (a) 7 (b) 3 (c) 1 (d) 9
12. 12.Which of these numbers is divisible by 8? (a) 4810 (b) 4808 (c) 4805 (d) 4820
13. 13.What is the units digit of 2^57? (a) 6 (b) 8 (c) 2 (d) 4
14. 14.What is the smallest number you must add to 115 to make it divisible by 4? (a) 3 (b) 2 (c) 1
15. 15.What is the units digit of 8^31? (a) 4 (b) 8 (c) 2 (d) 6
16. 16.What is the units digit of 3^34? (a) 7 (b) 9 (c) 1 (d) 3
17. 17.What is the remainder when 501 is divided by 7? (a) 5 (b) 3 (c) 6 (d) 4
18. 18.What is the remainder when 461 is divided by 7? (a) 1 (b) 4 (c) 5 (d) 6
19. 19.Which of these numbers is divisible by 3? (a) 1679 (b) 1685 (c) 1680 (d) 1682
20. 20.Which of these numbers is divisible by 6? (a) 2436 (b) 2445 (c) 2441 (d) 2432
21. 21.Which of these numbers is divisible by 3? (a) 1313 (b) 1314 (c) 1319 (d) 1315
22. 22.What is the remainder when 307 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
23. 23.What is the remainder when 181 is divided by 7? (a) 5 (b) 6 (c) 4 (d) 1
24. 24.What is the remainder when 245 is divided by 12? (a) 5 (b) 4 (c) 7 (d) 6
25. 25.What is the smallest number you must add to 261 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
26. 26.What is the remainder when 545 is divided by 9? (a) 6 (b) 7 (c) 4 (d) 5
27. 27.What is the smallest number you must add to 244 to make it divisible by 11? (a) 8 (b) 10 (c) 2 (d) 9
28. 28.What is the remainder when 849 is divided by 12? (a) 3 (b) 8 (c) 9 (d) 10
29. 29.What is the smallest number you must add to 379 to make it divisible by 5? (a) 1 (b) 2 (c) 3 (d) 4
30. 30.Which of these numbers is divisible by 4? (a) 2696 (b) 2701 (c) 2697 (d) 2694
31. 31.What is the units digit of 4^41? (a) 3 (b) 5 (c) 6 (d) 4
32. 32.What is the smallest number you must add to 93 to make it divisible by 4? (a) 1 (b) 3 (c) 2
33. 33.What is the remainder when 229 is divided by 6? (a) 1 (b) 0 (c) 5 (d) 2
34. 34.What is the smallest number you must add to 155 to make it divisible by 7? (a) 4 (b) 1 (c) 5 (d) 6
35. 35.What is the units digit of 3^50? (a) 1 (b) 9 (c) 3 (d) 7
36. 36.What is the units digit of 2^50? (a) 2 (b) 4 (c) 6 (d) 8
37. 37.Which of these numbers is divisible by 4? (a) 2418 (b) 2410 (c) 2414 (d) 2412
38. 38.What is the remainder when 431 is divided by 8? (a) 6 (b) 1 (c) 7 (d) 5
39. 39.Which of these numbers is divisible by 11? (a) 6113 (b) 6094 (c) 6092 (d) 6100
40. 40.What is the smallest number you must add to 79 to make it divisible by 4? (a) 1 (b) 2 (c) 3

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