Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 58 · 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 3^26? (a) 9 (b) 7 (c) 1 (d) 3
2. 2.Which of these numbers is divisible by 4? (a) 3191 (b) 3187 (c) 3194 (d) 3188
3. 3.What is the remainder when 98 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
4. 4.What is the remainder when 71 is divided by 5? (a) 2 (b) 4 (c) 0 (d) 1
5. 5.What is the smallest number you must add to 107 to make it divisible by 6? (a) 2 (b) 5 (c) 3 (d) 1
6. 6.Which of these numbers is divisible by 11? (a) 4413 (b) 4400 (c) 4390 (d) 4409
7. 7.What is the units digit of 2^40? (a) 6 (b) 8 (c) 4 (d) 2
8. 8.What is the remainder when 76 is divided by 5? (a) 2 (b) 0 (c) 4 (d) 1
9. 9.What is the units digit of 8^27? (a) 4 (b) 2 (c) 6 (d) 8
10. 10.Which of these numbers is divisible by 11? (a) 4575 (b) 4544 (c) 4558 (d) 4554
11. 11.What is the remainder when 333 is divided by 11? (a) 3 (b) 2 (c) 8 (d) 4
12. 12.Which of these numbers is divisible by 11? (a) 4677 (b) 4694 (c) 4698 (d) 4686
13. 13.What is the smallest number you must add to 131 to make it divisible by 8? (a) 6 (b) 3 (c) 4 (d) 5
14. 14.Which of these numbers is divisible by 4? (a) 678 (b) 672 (c) 674 (d) 670
15. 15.Which of these numbers is divisible by 3? (a) 2369 (b) 2371 (c) 2365 (d) 2367
16. 16.Which of these numbers is divisible by 9? (a) 8003 (b) 8010 (c) 8014 (d) 8021
17. 17.Which of these numbers is divisible by 8? (a) 1446 (b) 1448 (c) 1459 (d) 1455
18. 18.What is the remainder when 366 is divided by 11? (a) 4 (b) 8 (c) 3 (d) 2
19. 19.What is the remainder when 691 is divided by 9? (a) 8 (b) 7 (c) 2 (d) 6
20. 20.What is the remainder when 141 is divided by 7? (a) 6 (b) 2 (c) 0 (d) 1
21. 21.What is the units digit of 4^53? (a) 6 (b) 5 (c) 4 (d) 3
22. 22.What is the units digit of 8^11? (a) 4 (b) 2 (c) 8 (d) 6
23. 23.What is the smallest number you must add to 323 to make it divisible by 4? (a) 2 (b) 1 (c) 3
24. 24.What is the smallest number you must add to 184 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
25. 25.What is the smallest number you must add to 46 to make it divisible by 3? (a) 1 (b) 2
26. 26.What is the smallest number you must add to 135 to make it divisible by 7? (a) 4 (b) 6 (c) 2 (d) 5
27. 27.Which of these numbers is divisible by 3? (a) 1010 (b) 1006 (c) 1013 (d) 1008
28. 28.Which of these numbers is divisible by 8? (a) 366 (b) 360 (c) 357 (d) 373
29. 29.What is the units digit of 4^50? (a) 4 (b) 6 (c) 5 (d) 7
30. 30.What is the units digit of 7^21? (a) 9 (b) 1 (c) 7 (d) 3
31. 31.What is the units digit of 7^17? (a) 3 (b) 9 (c) 7 (d) 1
32. 32.What is the units digit of 9^15? (a) 8 (b) 7 (c) 1 (d) 9
33. 33.What is the remainder when 117 is divided by 5? (a) 4 (b) 3 (c) 2 (d) 1
34. 34.Which of these numbers is divisible by 3? (a) 1863 (b) 1864 (c) 1862 (d) 1868
35. 35.What is the units digit of 4^32? (a) 7 (b) 6 (c) 5 (d) 4
36. 36.What is the smallest number you must add to 104 to make it divisible by 3? (a) 1 (b) 2
37. 37.Which of these numbers is divisible by 3? (a) 2056 (b) 2060 (c) 2063 (d) 2058
38. 38.Which of these numbers is divisible by 9? (a) 4322 (b) 4331 (c) 4318 (d) 4320
39. 39.What is the smallest number you must add to 338 to make it divisible by 6? (a) 3 (b) 5 (c) 2 (d) 4
40. 40.What is the units digit of 7^60? (a) 3 (b) 1 (c) 7 (d) 9

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