Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 892 · 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 9? (a) 441 (b) 449 (c) 434 (d) 456
2. 2.What is the units digit of 2^19? (a) 8 (b) 6 (c) 4 (d) 2
3. 3.What is the smallest number you must add to 72 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
4. 4.What is the smallest number you must add to 550 to make it divisible by 9? (a) 6 (b) 8 (c) 7 (d) 1
5. 5.What is the smallest number you must add to 796 to make it divisible by 11? (a) 6 (b) 8 (c) 4 (d) 7
6. 6.What is the remainder when 154 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
7. 7.What is the smallest number you must add to 65 to make it divisible by 3? (a) 1 (b) 2
8. 8.What is the units digit of 7^10? (a) 9 (b) 7 (c) 3 (d) 1
9. 9.What is the remainder when 146 is divided by 11? (a) 2 (b) 3 (c) 8 (d) 4
10. 10.What is the remainder when 184 is divided by 3? (a) 0 (b) 1 (c) 2
11. 11.What is the remainder when 189 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
12. 12.Which of these numbers is divisible by 9? (a) 4727 (b) 4723 (c) 4713 (d) 4716
13. 13.What is the units digit of 7^22? (a) 3 (b) 7 (c) 9 (d) 1
14. 14.What is the remainder when 921 is divided by 12? (a) 8 (b) 9 (c) 10 (d) 3
15. 15.What is the smallest number you must add to 292 to make it divisible by 7? (a) 5 (b) 2 (c) 1 (d) 3
16. 16.What is the units digit of 7^42? (a) 7 (b) 9 (c) 3 (d) 1
17. 17.What is the smallest number you must add to 332 to make it divisible by 5? (a) 4 (b) 1 (c) 3 (d) 2
18. 18.Which of these numbers is divisible by 8? (a) 2875 (b) 2862 (c) 2866 (d) 2864
19. 19.What is the remainder when 618 is divided by 9? (a) 7 (b) 6 (c) 3 (d) 5
20. 20.What is the remainder when 432 is divided by 10? (a) 8 (b) 3 (c) 1 (d) 2
21. 21.What is the smallest number you must add to 118 to make it divisible by 11? (a) 2 (b) 3 (c) 4 (d) 8
22. 22.Which of these numbers is divisible by 8? (a) 3170 (b) 3160 (c) 3157 (d) 3165
23. 23.What is the remainder when 441 is divided by 8? (a) 0 (b) 7 (c) 1 (d) 2
24. 24.Which of these numbers is divisible by 9? (a) 3335 (b) 3330 (c) 3328 (d) 3340
25. 25.What is the remainder when 214 is divided by 7? (a) 3 (b) 4 (c) 6 (d) 5
26. 26.What is the units digit of 9^52? (a) 9 (b) 2 (c) 1 (d) 0
27. 27.Which of these numbers is divisible by 8? (a) 1512 (b) 1511 (c) 1526 (d) 1514
28. 28.What is the remainder when 179 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
29. 29.What is the remainder when 269 is divided by 9? (a) 8 (b) 7 (c) 6 (d) 1
30. 30.What is the units digit of 3^32? (a) 7 (b) 1 (c) 3 (d) 9
31. 31.What is the units digit of 4^53? (a) 5 (b) 3 (c) 4 (d) 6
32. 32.What is the remainder when 266 is divided by 6? (a) 1 (b) 4 (c) 2 (d) 3
33. 33.What is the remainder when 112 is divided by 10? (a) 8 (b) 2 (c) 3 (d) 1
34. 34.What is the smallest number you must add to 559 to make it divisible by 11? (a) 9 (b) 3 (c) 2 (d) 1
35. 35.Which of these numbers is divisible by 8? (a) 3993 (b) 4004 (c) 3985 (d) 3992
36. 36.What is the units digit of 2^50? (a) 8 (b) 2 (c) 4 (d) 6
37. 37.Which of these numbers is divisible by 3? (a) 1306 (b) 1303 (c) 1309 (d) 1305
38. 38.Which of these numbers is divisible by 9? (a) 752 (b) 747 (c) 745 (d) 757
39. 39.What is the smallest number you must add to 151 to make it divisible by 11? (a) 2 (b) 4 (c) 3 (d) 8
40. 40.What is the smallest number you must add to 743 to make it divisible by 11? (a) 6 (b) 4 (c) 5 (d) 7

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