Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 33 · 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 6? (a) 838 (b) 828 (c) 823 (d) 829
2. 2.What is the smallest number you must add to 125 to make it divisible by 4? (a) 1 (b) 2 (c) 3
3. 3.What is the smallest number you must add to 464 to make it divisible by 7? (a) 5 (b) 6 (c) 4 (d) 2
4. 4.What is the units digit of 2^59? (a) 4 (b) 6 (c) 2 (d) 8
5. 5.Which of these numbers is divisible by 11? (a) 8939 (b) 8921 (c) 8920 (d) 8927
6. 6.Which of these numbers is divisible by 11? (a) 1584 (b) 1585 (c) 1601 (d) 1580
7. 7.What is the remainder when 675 is divided by 12? (a) 3 (b) 9 (c) 2 (d) 4
8. 8.Which of these numbers is divisible by 6? (a) 4411 (b) 4416 (c) 4426 (d) 4419
9. 9.Which of these numbers is divisible by 8? (a) 2636 (b) 2622 (c) 2626 (d) 2624
10. 10.What is the units digit of 3^12? (a) 9 (b) 7 (c) 3 (d) 1
11. 11.What is the smallest number you must add to 174 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
12. 12.What is the smallest number you must add to 103 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
13. 13.What is the units digit of 7^42? (a) 7 (b) 1 (c) 9 (d) 3
14. 14.Which of these numbers is divisible by 11? (a) 5977 (b) 5970 (c) 5988 (d) 5973
15. 15.What is the smallest number you must add to 244 to make it divisible by 8? (a) 3 (b) 4 (c) 5 (d) 6
16. 16.What is the units digit of 4^24? (a) 7 (b) 6 (c) 5 (d) 4
17. 17.What is the remainder when 446 is divided by 12? (a) 1 (b) 2 (c) 3 (d) 10
18. 18.What is the units digit of 4^27? (a) 5 (b) 3 (c) 4 (d) 6
19. 19.What is the units digit of 2^24? (a) 6 (b) 4 (c) 8 (d) 2
20. 20.What is the smallest number you must add to 263 to make it divisible by 6? (a) 5 (b) 1 (c) 3 (d) 2
21. 21.Which of these numbers is divisible by 8? (a) 449 (b) 445 (c) 448 (d) 457
22. 22.What is the units digit of 3^15? (a) 1 (b) 9 (c) 3 (d) 7
23. 23.What is the smallest number you must add to 505 to make it divisible by 8? (a) 7 (b) 5 (c) 1 (d) 6
24. 24.What is the remainder when 122 is divided by 11? (a) 10 (b) 0 (c) 1 (d) 2
25. 25.What is the remainder when 89 is divided by 8? (a) 2 (b) 1 (c) 7 (d) 0
26. 26.What is the smallest number you must add to 170 to make it divisible by 7? (a) 4 (b) 6 (c) 2 (d) 5
27. 27.What is the smallest number you must add to 166 to make it divisible by 3? (a) 1 (b) 2
28. 28.Which of these numbers is divisible by 4? (a) 1269 (b) 1264 (c) 1263 (d) 1266
29. 29.What is the smallest number you must add to 306 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
30. 30.What is the remainder when 61 is divided by 3? (a) 1 (b) 0 (c) 2
31. 31.What is the smallest number you must add to 163 to make it divisible by 12? (a) 4 (b) 5 (c) 6 (d) 7
32. 32.What is the units digit of 3^44? (a) 7 (b) 9 (c) 1 (d) 3
33. 33.What is the remainder when 138 is divided by 11? (a) 5 (b) 7 (c) 8 (d) 6
34. 34.What is the units digit of 2^31? (a) 4 (b) 2 (c) 6 (d) 8
35. 35.What is the smallest number you must add to 388 to make it divisible by 6? (a) 3 (b) 1 (c) 4 (d) 2
36. 36.What is the smallest number you must add to 261 to make it divisible by 10? (a) 8 (b) 9 (c) 1 (d) 7
37. 37.What is the remainder when 340 is divided by 8? (a) 3 (b) 4 (c) 5 (d) 6
38. 38.What is the smallest number you must add to 289 to make it divisible by 6? (a) 1 (b) 5 (c) 3 (d) 4
39. 39.What is the units digit of 7^19? (a) 9 (b) 3 (c) 1 (d) 7
40. 40.Which of these numbers is divisible by 9? (a) 3298 (b) 3306 (c) 3294 (d) 3287

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