Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 924 · 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 810 to make it divisible by 11? (a) 7 (b) 5 (c) 3 (d) 4
2. 2.What is the smallest number you must add to 283 to make it divisible by 4? (a) 3 (b) 2 (c) 1
3. 3.What is the remainder when 363 is divided by 10? (a) 3 (b) 4 (c) 2 (d) 7
4. 4.What is the units digit of 8^37? (a) 4 (b) 6 (c) 8 (d) 2
5. 5.What is the units digit of 4^20? (a) 6 (b) 4 (c) 5 (d) 7
6. 6.What is the smallest number you must add to 165 to make it divisible by 4? (a) 3 (b) 1 (c) 2
7. 7.What is the smallest number you must add to 131 to make it divisible by 6? (a) 1 (b) 5 (c) 2 (d) 3
8. 8.What is the remainder when 477 is divided by 8? (a) 3 (b) 6 (c) 4 (d) 5
9. 9.Which of these numbers is divisible by 11? (a) 9181 (b) 9169 (c) 9163 (d) 9156
10. 10.What is the smallest number you must add to 488 to make it divisible by 9? (a) 8 (b) 2 (c) 7 (d) 6
11. 11.Which of these numbers is divisible by 4? (a) 1074 (b) 1068 (c) 1065 (d) 1070
12. 12.Which of these numbers is divisible by 8? (a) 3624 (b) 3622 (c) 3626 (d) 3636
13. 13.What is the units digit of 9^46? (a) 9 (b) 1 (c) 0 (d) 2
14. 14.Which of these numbers is divisible by 9? (a) 4537 (b) 4545 (c) 4553 (d) 4560
15. 15.Which of these numbers is divisible by 11? (a) 1057 (b) 1070 (c) 1056 (d) 1052
16. 16.Which of these numbers is divisible by 4? (a) 1868 (b) 1865 (c) 1869 (d) 1875
17. 17.What is the units digit of 9^59? (a) 1 (b) 7 (c) 8 (d) 9
18. 18.Which of these numbers is divisible by 9? (a) 1044 (b) 1040 (c) 1054 (d) 1046
19. 19.Which of these numbers is divisible by 4? (a) 500 (b) 499 (c) 505 (d) 502
20. 20.Which of these numbers is divisible by 8? (a) 2560 (b) 2556 (c) 2570 (d) 2562
21. 21.What is the smallest number you must add to 574 to make it divisible by 8? (a) 2 (b) 1 (c) 3 (d) 6
22. 22.Which of these numbers is divisible by 4? (a) 1106 (b) 1100 (c) 1103 (d) 1099
23. 23.What is the units digit of 7^13? (a) 9 (b) 7 (c) 1 (d) 3
24. 24.What is the remainder when 359 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
25. 25.What is the smallest number you must add to 258 to make it divisible by 9? (a) 2 (b) 3 (c) 4 (d) 6
26. 26.What is the remainder when 122 is divided by 9? (a) 5 (b) 6 (c) 7 (d) 4
27. 27.Which of these numbers is divisible by 3? (a) 592 (b) 596 (c) 598 (d) 594
28. 28.Which of these numbers is divisible by 6? (a) 2218 (b) 2214 (c) 2209 (d) 2225
29. 29.What is the units digit of 4^18? (a) 5 (b) 4 (c) 6 (d) 7
30. 30.What is the units digit of 7^32? (a) 9 (b) 3 (c) 7 (d) 1
31. 31.What is the smallest number you must add to 67 to make it divisible by 3? (a) 2 (b) 1
32. 32.What is the remainder when 225 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
33. 33.What is the units digit of 7^18? (a) 9 (b) 3 (c) 1 (d) 7
34. 34.What is the smallest number you must add to 161 to make it divisible by 3? (a) 1 (b) 2
35. 35.Which of these numbers is divisible by 8? (a) 2072 (b) 2084 (c) 2078 (d) 2066
36. 36.What is the units digit of 2^50? (a) 8 (b) 2 (c) 6 (d) 4
37. 37.What is the remainder when 268 is divided by 7? (a) 2 (b) 1 (c) 5 (d) 3
38. 38.Which of these numbers is divisible by 6? (a) 2232 (b) 2233 (c) 2242 (d) 2228
39. 39.What is the remainder when 442 is divided by 6? (a) 2 (b) 5 (c) 4 (d) 3
40. 40.What is the units digit of 8^14? (a) 4 (b) 2 (c) 8 (d) 6

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