Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 952 · 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 527 is divided by 8? (a) 7 (b) 1 (c) 5 (d) 6
2. 2.Which of these numbers is divisible by 8? (a) 3343 (b) 3344 (c) 3353 (d) 3346
3. 3.What is the units digit of 4^20? (a) 7 (b) 4 (c) 5 (d) 6
4. 4.Which of these numbers is divisible by 11? (a) 9301 (b) 9277 (c) 9284 (d) 9293
5. 5.What is the smallest number you must add to 133 to make it divisible by 3? (a) 1 (b) 2
6. 6.Which of these numbers is divisible by 8? (a) 6212 (b) 6218 (c) 6228 (d) 6216
7. 7.What is the remainder when 223 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
8. 8.What is the units digit of 3^45? (a) 1 (b) 7 (c) 3 (d) 9
9. 9.Which of these numbers is divisible by 4? (a) 419 (b) 409 (c) 413 (d) 412
10. 10.What is the smallest number you must add to 311 to make it divisible by 4? (a) 3 (b) 2 (c) 1
11. 11.What is the units digit of 2^38? (a) 2 (b) 6 (c) 8 (d) 4
12. 12.Which of these numbers is divisible by 6? (a) 5379 (b) 5391 (c) 5382 (d) 5384
13. 13.What is the units digit of 7^44? (a) 3 (b) 9 (c) 1 (d) 7
14. 14.What is the smallest number you must add to 298 to make it divisible by 4? (a) 3 (b) 2 (c) 1
15. 15.What is the remainder when 628 is divided by 9? (a) 2 (b) 6 (c) 8 (d) 7
16. 16.What is the smallest number you must add to 224 to make it divisible by 11? (a) 7 (b) 6 (c) 8 (d) 4
17. 17.Which of these numbers is divisible by 8? (a) 4104 (b) 4107 (c) 4103 (d) 4119
18. 18.What is the units digit of 3^40? (a) 1 (b) 9 (c) 3 (d) 7
19. 19.What is the smallest number you must add to 568 to make it divisible by 11? (a) 4 (b) 5 (c) 3 (d) 7
20. 20.Which of these numbers is divisible by 11? (a) 9559 (b) 9573 (c) 9565 (d) 9553
21. 21.What is the remainder when 645 is divided by 10? (a) 6 (b) 4 (c) 7 (d) 5
22. 22.What is the units digit of 4^32? (a) 6 (b) 4 (c) 5 (d) 7
23. 23.What is the smallest number you must add to 478 to make it divisible by 10? (a) 3 (b) 1 (c) 8 (d) 2
24. 24.What is the remainder when 285 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
25. 25.What is the smallest number you must add to 383 to make it divisible by 11? (a) 2 (b) 3 (c) 1 (d) 9
26. 26.What is the smallest number you must add to 387 to make it divisible by 7? (a) 5 (b) 6 (c) 2 (d) 4
27. 27.What is the smallest number you must add to 53 to make it divisible by 4? (a) 2 (b) 1 (c) 3
28. 28.What is the remainder when 152 is divided by 3? (a) 2 (b) 1 (c) 0
29. 29.What is the remainder when 142 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
30. 30.What is the smallest number you must add to 127 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
31. 31.What is the smallest number you must add to 269 to make it divisible by 10? (a) 1 (b) 3 (c) 2 (d) 9
32. 32.What is the units digit of 3^58? (a) 3 (b) 1 (c) 7 (d) 9
33. 33.What is the units digit of 9^54? (a) 9 (b) 2 (c) 0 (d) 1
34. 34.What is the units digit of 2^59? (a) 8 (b) 6 (c) 4 (d) 2
35. 35.Which of these numbers is divisible by 8? (a) 5165 (b) 5172 (c) 5154 (d) 5160
36. 36.What is the remainder when 136 is divided by 5? (a) 2 (b) 0 (c) 1 (d) 4
37. 37.What is the units digit of 8^56? (a) 2 (b) 4 (c) 6 (d) 8
38. 38.What is the smallest number you must add to 200 to make it divisible by 3? (a) 1 (b) 2
39. 39.Which of these numbers is divisible by 4? (a) 2358 (b) 2360 (c) 2362 (d) 2366
40. 40.Which of these numbers is divisible by 3? (a) 569 (b) 571 (c) 565 (d) 567

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