Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 101 · 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 smallest number you must add to 181 to make it divisible by 4? (a) 2 (b) 3 (c) 1
2. 2.Which of these numbers is divisible by 9? (a) 6884 (b) 6867 (c) 6874 (d) 6866
3. 3.What is the units digit of 4^42? (a) 6 (b) 4 (c) 5 (d) 7
4. 4.What is the remainder when 459 is divided by 10? (a) 1 (b) 8 (c) 7 (d) 9
5. 5.What is the smallest number you must add to 164 to make it divisible by 6? (a) 2 (b) 3 (c) 4 (d) 5
6. 6.Which of these numbers is divisible by 9? (a) 540 (b) 533 (c) 546 (d) 552
7. 7.Which of these numbers is divisible by 9? (a) 394 (b) 378 (c) 374 (d) 384
8. 8.What is the smallest number you must add to 442 to make it divisible by 12? (a) 1 (b) 3 (c) 2 (d) 10
9. 9.What is the units digit of 8^37? (a) 6 (b) 4 (c) 8 (d) 2
10. 10.What is the smallest number you must add to 257 to make it divisible by 6? (a) 5 (b) 1 (c) 2 (d) 3
11. 11.What is the units digit of 4^28? (a) 5 (b) 4 (c) 7 (d) 6
12. 12.What is the remainder when 163 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
13. 13.Which of these numbers is divisible by 8? (a) 7109 (b) 7101 (c) 7115 (d) 7104
14. 14.Which of these numbers is divisible by 8? (a) 376 (b) 371 (c) 380 (d) 389
15. 15.What is the smallest number you must add to 202 to make it divisible by 4? (a) 2 (b) 3 (c) 1
16. 16.What is the units digit of 7^20? (a) 7 (b) 1 (c) 9 (d) 3
17. 17.What is the smallest number you must add to 114 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
18. 18.What is the units digit of 8^11? (a) 6 (b) 4 (c) 2 (d) 8
19. 19.What is the remainder when 181 is divided by 3? (a) 2 (b) 1 (c) 0
20. 20.What is the smallest number you must add to 175 to make it divisible by 10? (a) 6 (b) 7 (c) 5 (d) 4
21. 21.What is the units digit of 2^31? (a) 2 (b) 6 (c) 8 (d) 4
22. 22.Which of these numbers is divisible by 8? (a) 3917 (b) 3926 (c) 3912 (d) 3907
23. 23.What is the remainder when 685 is divided by 10? (a) 7 (b) 5 (c) 4 (d) 6
24. 24.What is the remainder when 514 is divided by 10? (a) 4 (b) 5 (c) 3 (d) 6
25. 25.Which of these numbers is divisible by 8? (a) 3482 (b) 3479 (c) 3480 (d) 3495
26. 26.What is the smallest number you must add to 637 to make it divisible by 9? (a) 2 (b) 7 (c) 3 (d) 1
27. 27.What is the remainder when 193 is divided by 7? (a) 5 (b) 3 (c) 6 (d) 4
28. 28.What is the remainder when 428 is divided by 10? (a) 9 (b) 7 (c) 2 (d) 8
29. 29.Which of these numbers is divisible by 3? (a) 1120 (b) 1124 (c) 1119 (d) 1118
30. 30.Which of these numbers is divisible by 4? (a) 599 (b) 592 (c) 593 (d) 591
31. 31.Which of these numbers is divisible by 11? (a) 9476 (b) 9464 (c) 9491 (d) 9471
32. 32.What is the units digit of 3^20? (a) 7 (b) 9 (c) 3 (d) 1
33. 33.Which of these numbers is divisible by 11? (a) 4667 (b) 4675 (c) 4678 (d) 4690
34. 34.What is the smallest number you must add to 167 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
35. 35.What is the units digit of 7^25? (a) 1 (b) 7 (c) 3 (d) 9
36. 36.What is the smallest number you must add to 481 to make it divisible by 6? (a) 4 (b) 5 (c) 3 (d) 1
37. 37.What is the smallest number you must add to 111 to make it divisible by 10? (a) 8 (b) 1 (c) 7 (d) 9
38. 38.Which of these numbers is divisible by 9? (a) 7018 (b) 7027 (c) 7035 (d) 7020
39. 39.What is the remainder when 81 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
40. 40.What is the units digit of 3^11? (a) 1 (b) 3 (c) 7 (d) 9

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