Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 89 · 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 remainder when 352 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
2. 2.What is the smallest number you must add to 302 to make it divisible by 7? (a) 1 (b) 5 (c) 6 (d) 4
3. 3.What is the smallest number you must add to 154 to make it divisible by 3? (a) 1 (b) 2
4. 4.What is the units digit of 3^27? (a) 7 (b) 9 (c) 1 (d) 3
5. 5.What is the smallest number you must add to 71 to make it divisible by 3? (a) 2 (b) 1
6. 6.Which of these numbers is divisible by 3? (a) 2127 (b) 2132 (c) 2129 (d) 2126
7. 7.What is the remainder when 377 is divided by 8? (a) 1 (b) 7 (c) 2 (d) 0
8. 8.What is the smallest number you must add to 169 to make it divisible by 8? (a) 6 (b) 7 (c) 5 (d) 1
9. 9.What is the units digit of 2^31? (a) 8 (b) 2 (c) 4 (d) 6
10. 10.What is the units digit of 8^11? (a) 2 (b) 8 (c) 4 (d) 6
11. 11.What is the smallest number you must add to 703 to make it divisible by 9? (a) 8 (b) 7 (c) 1 (d) 6
12. 12.What is the units digit of 2^58? (a) 2 (b) 8 (c) 6 (d) 4
13. 13.What is the units digit of 8^17? (a) 4 (b) 8 (c) 2 (d) 6
14. 14.What is the smallest number you must add to 271 to make it divisible by 6? (a) 3 (b) 5 (c) 1 (d) 4
15. 15.What is the units digit of 8^43? (a) 4 (b) 6 (c) 8 (d) 2
16. 16.What is the remainder when 353 is divided by 10? (a) 3 (b) 2 (c) 7 (d) 4
17. 17.What is the remainder when 253 is divided by 9? (a) 2 (b) 0 (c) 1 (d) 8
18. 18.What is the smallest number you must add to 955 to make it divisible by 12? (a) 5 (b) 6 (c) 4 (d) 7
19. 19.What is the smallest number you must add to 239 to make it divisible by 12? (a) 2 (b) 3 (c) 11 (d) 1
20. 20.Which of these numbers is divisible by 11? (a) 5551 (b) 5563 (c) 5544 (d) 5543
21. 21.What is the remainder when 786 is divided by 12? (a) 8 (b) 6 (c) 7 (d) 5
22. 22.What is the units digit of 4^33? (a) 5 (b) 4 (c) 3 (d) 6
23. 23.What is the units digit of 4^26? (a) 4 (b) 5 (c) 6 (d) 7
24. 24.What is the units digit of 3^58? (a) 1 (b) 9 (c) 3 (d) 7
25. 25.Which of these numbers is divisible by 6? (a) 1294 (b) 1303 (c) 1296 (d) 1297
26. 26.What is the smallest number you must add to 384 to make it divisible by 11? (a) 10 (b) 2 (c) 1 (d) 3
27. 27.What is the units digit of 8^16? (a) 4 (b) 2 (c) 6 (d) 8
28. 28.What is the smallest number you must add to 966 to make it divisible by 12? (a) 6 (b) 7 (c) 5 (d) 8
29. 29.Which of these numbers is divisible by 8? (a) 3085 (b) 3065 (c) 3072 (d) 3078
30. 30.Which of these numbers is divisible by 8? (a) 5277 (b) 5289 (c) 5285 (d) 5280
31. 31.What is the smallest number you must add to 326 to make it divisible by 9? (a) 8 (b) 7 (c) 6 (d) 2
32. 32.Which of these numbers is divisible by 9? (a) 498 (b) 494 (c) 495 (d) 507
33. 33.What is the remainder when 348 is divided by 5? (a) 3 (b) 1 (c) 2 (d) 4
34. 34.Which of these numbers is divisible by 6? (a) 5030 (b) 5028 (c) 5038 (d) 5024
35. 35.What is the units digit of 8^44? (a) 6 (b) 2 (c) 8 (d) 4
36. 36.What is the remainder when 139 is divided by 9? (a) 5 (b) 6 (c) 4 (d) 3
37. 37.What is the remainder when 551 is divided by 9? (a) 3 (b) 7 (c) 1 (d) 2
38. 38.What is the units digit of 7^59? (a) 9 (b) 1 (c) 3 (d) 7
39. 39.What is the remainder when 229 is divided by 9? (a) 4 (b) 6 (c) 5 (d) 3
40. 40.What is the smallest number you must add to 61 to make it divisible by 4? (a) 2 (b) 1 (c) 3

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad