Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 857 · 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 4? (a) 2038 (b) 2033 (c) 2032 (d) 2029
2. 2.What is the units digit of 7^18? (a) 3 (b) 9 (c) 7 (d) 1
3. 3.Which of these numbers is divisible by 4? (a) 3016 (b) 3014 (c) 3023 (d) 3018
4. 4.Which of these numbers is divisible by 3? (a) 2036 (b) 2037 (c) 2039 (d) 2042
5. 5.What is the units digit of 4^10? (a) 7 (b) 5 (c) 4 (d) 6
6. 6.Which of these numbers is divisible by 3? (a) 346 (b) 348 (c) 352 (d) 350
7. 7.What is the remainder when 194 is divided by 11? (a) 8 (b) 6 (c) 7 (d) 4
8. 8.What is the remainder when 527 is divided by 10? (a) 3 (b) 8 (c) 7 (d) 6
9. 9.What is the units digit of 2^12? (a) 4 (b) 8 (c) 6 (d) 2
10. 10.Which of these numbers is divisible by 3? (a) 835 (b) 831 (c) 833 (d) 829
11. 11.What is the remainder when 359 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
12. 12.Which of these numbers is divisible by 3? (a) 403 (b) 409 (c) 407 (d) 405
13. 13.What is the units digit of 2^46? (a) 8 (b) 6 (c) 2 (d) 4
14. 14.What is the units digit of 4^17? (a) 3 (b) 4 (c) 6 (d) 5
15. 15.What is the smallest number you must add to 53 to make it divisible by 3? (a) 1 (b) 2
16. 16.What is the remainder when 691 is divided by 12? (a) 7 (b) 6 (c) 5 (d) 8
17. 17.What is the remainder when 429 is divided by 12? (a) 3 (b) 8 (c) 10 (d) 9
18. 18.What is the smallest number you must add to 387 to make it divisible by 8? (a) 6 (b) 5 (c) 4 (d) 3
19. 19.Which of these numbers is divisible by 4? (a) 1066 (b) 1058 (c) 1061 (d) 1060
20. 20.Which of these numbers is divisible by 6? (a) 535 (b) 541 (c) 533 (d) 534
21. 21.What is the smallest number you must add to 146 to make it divisible by 3? (a) 1 (b) 2
22. 22.What is the units digit of 4^23? (a) 6 (b) 5 (c) 4 (d) 3
23. 23.What is the remainder when 151 is divided by 9? (a) 7 (b) 2 (c) 6 (d) 8
24. 24.What is the remainder when 178 is divided by 8? (a) 2 (b) 1 (c) 6 (d) 3
25. 25.Which of these numbers is divisible by 11? (a) 6798 (b) 6812 (c) 6790 (d) 6806
26. 26.What is the units digit of 4^30? (a) 5 (b) 7 (c) 6 (d) 4
27. 27.Which of these numbers is divisible by 9? (a) 6381 (b) 6382 (c) 6378 (d) 6393
28. 28.What is the units digit of 3^49? (a) 9 (b) 7 (c) 3 (d) 1
29. 29.Which of these numbers is divisible by 4? (a) 1620 (b) 1619 (c) 1622 (d) 1625
30. 30.Which of these numbers is divisible by 6? (a) 1410 (b) 1405 (c) 1420 (d) 1412
31. 31.What is the smallest number you must add to 118 to make it divisible by 7? (a) 2 (b) 6 (c) 3 (d) 1
32. 32.Which of these numbers is divisible by 11? (a) 3290 (b) 3281 (c) 3289 (d) 3307
33. 33.Which of these numbers is divisible by 3? (a) 274 (b) 270 (c) 271 (d) 269
34. 34.What is the remainder when 232 is divided by 6? (a) 2 (b) 3 (c) 4 (d) 5
35. 35.What is the remainder when 83 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
36. 36.What is the remainder when 182 is divided by 6? (a) 1 (b) 4 (c) 3 (d) 2
37. 37.What is the smallest number you must add to 107 to make it divisible by 4? (a) 3 (b) 1 (c) 2
38. 38.What is the remainder when 494 is divided by 8? (a) 5 (b) 7 (c) 2 (d) 6
39. 39.What is the units digit of 7^23? (a) 7 (b) 9 (c) 3 (d) 1
40. 40.Which of these numbers is divisible by 11? (a) 1032 (b) 1003 (c) 1022 (d) 1012

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