Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 260 · 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 745 is divided by 12? (a) 0 (b) 2 (c) 1 (d) 11
2. 2.Which of these numbers is divisible by 8? (a) 500 (b) 486 (c) 488 (d) 493
3. 3.What is the units digit of 4^10? (a) 7 (b) 6 (c) 4 (d) 5
4. 4.Which of these numbers is divisible by 6? (a) 3261 (b) 3258 (c) 3267 (d) 3254
5. 5.What is the units digit of 7^16? (a) 3 (b) 1 (c) 9 (d) 7
6. 6.What is the units digit of 8^30? (a) 2 (b) 6 (c) 4 (d) 8
7. 7.What is the smallest number you must add to 161 to make it divisible by 4? (a) 1 (b) 3 (c) 2
8. 8.Which of these numbers is divisible by 3? (a) 1616 (b) 1613 (c) 1609 (d) 1611
9. 9.Which of these numbers is divisible by 9? (a) 5350 (b) 5344 (c) 5334 (d) 5337
10. 10.What is the smallest number you must add to 467 to make it divisible by 7? (a) 1 (b) 3 (c) 5 (d) 2
11. 11.What is the units digit of 3^51? (a) 1 (b) 3 (c) 7 (d) 9
12. 12.Which of these numbers is divisible by 6? (a) 885 (b) 896 (c) 892 (d) 888
13. 13.What is the remainder when 307 is divided by 8? (a) 2 (b) 5 (c) 4 (d) 3
14. 14.What is the remainder when 522 is divided by 8? (a) 2 (b) 6 (c) 3 (d) 1
15. 15.What is the units digit of 7^37? (a) 7 (b) 1 (c) 9 (d) 3
16. 16.What is the smallest number you must add to 102 to make it divisible by 5? (a) 3 (b) 1 (c) 4 (d) 2
17. 17.What is the smallest number you must add to 129 to make it divisible by 8? (a) 1 (b) 6 (c) 5 (d) 7
18. 18.What is the smallest number you must add to 227 to make it divisible by 3? (a) 1 (b) 2
19. 19.What is the units digit of 8^18? (a) 6 (b) 8 (c) 4 (d) 2
20. 20.Which of these numbers is divisible by 3? (a) 1980 (b) 1985 (c) 1978 (d) 1982
21. 21.Which of these numbers is divisible by 9? (a) 1117 (b) 1101 (c) 1107 (d) 1113
22. 22.What is the smallest number you must add to 91 to make it divisible by 6? (a) 1 (b) 4 (c) 3 (d) 5
23. 23.What is the remainder when 661 is divided by 10? (a) 0 (b) 9 (c) 2 (d) 1
24. 24.What is the smallest number you must add to 141 to make it divisible by 6? (a) 4 (b) 5 (c) 2 (d) 3
25. 25.What is the remainder when 194 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
26. 26.What is the smallest number you must add to 563 to make it divisible by 8? (a) 5 (b) 4 (c) 3 (d) 6
27. 27.What is the smallest number you must add to 772 to make it divisible by 11? (a) 8 (b) 2 (c) 9 (d) 10
28. 28.What is the units digit of 3^25? (a) 7 (b) 3 (c) 9 (d) 1
29. 29.Which of these numbers is divisible by 8? (a) 3451 (b) 3458 (c) 3456 (d) 3469
30. 30.Which of these numbers is divisible by 11? (a) 2187 (b) 2206 (c) 2189 (d) 2190
31. 31.What is the units digit of 3^26? (a) 1 (b) 3 (c) 9 (d) 7
32. 32.What is the smallest number you must add to 613 to make it divisible by 8? (a) 2 (b) 4 (c) 3 (d) 5
33. 33.What is the units digit of 4^16? (a) 6 (b) 5 (c) 4 (d) 7
34. 34.Which of these numbers is divisible by 9? (a) 2093 (b) 2101 (c) 2097 (d) 2114
35. 35.What is the smallest number you must add to 349 to make it divisible by 10? (a) 2 (b) 3 (c) 1 (d) 9
36. 36.What is the smallest number you must add to 137 to make it divisible by 7? (a) 2 (b) 5 (c) 3 (d) 4
37. 37.What is the remainder when 545 is divided by 8? (a) 7 (b) 0 (c) 2 (d) 1
38. 38.What is the remainder when 845 is divided by 12? (a) 6 (b) 7 (c) 4 (d) 5
39. 39.What is the remainder when 218 is divided by 4? (a) 2 (b) 1 (c) 3 (d) 0
40. 40.What is the units digit of 2^44? (a) 4 (b) 2 (c) 6 (d) 8

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