Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 873 · 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) 2528 (b) 2535 (c) 2527 (d) 2530
2. 2.What is the remainder when 656 is divided by 10? (a) 4 (b) 6 (c) 5 (d) 7
3. 3.What is the smallest number you must add to 376 to make it divisible by 6? (a) 1 (b) 2 (c) 3 (d) 4
4. 4.What is the remainder when 65 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 5
5. 5.What is the remainder when 429 is divided by 9? (a) 7 (b) 5 (c) 3 (d) 6
6. 6.Which of these numbers is divisible by 3? (a) 869 (b) 874 (c) 870 (d) 872
7. 7.Which of these numbers is divisible by 11? (a) 1063 (b) 1044 (c) 1046 (d) 1045
8. 8.What is the smallest number you must add to 247 to make it divisible by 10? (a) 7 (b) 4 (c) 2 (d) 3
9. 9.What is the units digit of 3^59? (a) 1 (b) 3 (c) 7 (d) 9
10. 10.What is the remainder when 371 is divided by 10? (a) 1 (b) 9 (c) 2 (d) 0
11. 11.What is the remainder when 345 is divided by 11? (a) 7 (b) 4 (c) 3 (d) 5
12. 12.What is the units digit of 2^54? (a) 6 (b) 2 (c) 4 (d) 8
13. 13.Which of these numbers is divisible by 3? (a) 590 (b) 588 (c) 586 (d) 592
14. 14.What is the units digit of 8^51? (a) 4 (b) 8 (c) 6 (d) 2
15. 15.What is the smallest number you must add to 154 to make it divisible by 9? (a) 1 (b) 8 (c) 6 (d) 7
16. 16.What is the smallest number you must add to 232 to make it divisible by 3? (a) 2 (b) 1
17. 17.Which of these numbers is divisible by 6? (a) 2630 (b) 2626 (c) 2636 (d) 2628
18. 18.What is the remainder when 239 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
19. 19.What is the units digit of 7^16? (a) 3 (b) 1 (c) 9 (d) 7
20. 20.Which of these numbers is divisible by 6? (a) 258 (b) 261 (c) 265 (d) 257
21. 21.What is the smallest number you must add to 119 to make it divisible by 3? (a) 2 (b) 1
22. 22.Which of these numbers is divisible by 6? (a) 1395 (b) 1409 (c) 1398 (d) 1400
23. 23.What is the remainder when 628 is divided by 11? (a) 0 (b) 1 (c) 2 (d) 10
24. 24.Which of these numbers is divisible by 9? (a) 7847 (b) 7839 (c) 7853 (d) 7831
25. 25.What is the remainder when 304 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 3
26. 26.What is the remainder when 694 is divided by 11? (a) 0 (b) 2 (c) 1 (d) 10
27. 27.Which of these numbers is divisible by 9? (a) 3393 (b) 3391 (c) 3410 (d) 3400
28. 28.Which of these numbers is divisible by 4? (a) 2137 (b) 2133 (c) 2143 (d) 2136
29. 29.What is the remainder when 133 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
30. 30.What is the units digit of 9^50? (a) 0 (b) 1 (c) 2 (d) 9
31. 31.What is the remainder when 67 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
32. 32.Which of these numbers is divisible by 4? (a) 1456 (b) 1457 (c) 1453 (d) 1462
33. 33.What is the smallest number you must add to 166 to make it divisible by 6? (a) 4 (b) 2 (c) 1 (d) 3
34. 34.What is the smallest number you must add to 94 to make it divisible by 9? (a) 7 (b) 5 (c) 4 (d) 6
35. 35.Which of these numbers is divisible by 8? (a) 406 (b) 408 (c) 423 (d) 415
36. 36.Which of these numbers is divisible by 3? (a) 2021 (b) 2019 (c) 2018 (d) 2023
37. 37.What is the units digit of 8^25? (a) 6 (b) 8 (c) 2 (d) 4
38. 38.What is the smallest number you must add to 301 to make it divisible by 6? (a) 1 (b) 3 (c) 4 (d) 5
39. 39.What is the units digit of 3^30? (a) 3 (b) 7 (c) 1 (d) 9
40. 40.What is the remainder when 164 is divided by 3? (a) 1 (b) 2 (c) 0

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