Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 425 · 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 366 is divided by 7? (a) 5 (b) 3 (c) 2 (d) 1
2. 2.Which of these numbers is divisible by 6? (a) 1612 (b) 1615 (c) 1608 (d) 1607
3. 3.What is the remainder when 231 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
4. 4.Which of these numbers is divisible by 3? (a) 2166 (b) 2171 (c) 2164 (d) 2168
5. 5.What is the units digit of 7^15? (a) 7 (b) 1 (c) 9 (d) 3
6. 6.What is the remainder when 643 is divided by 11? (a) 6 (b) 4 (c) 7 (d) 5
7. 7.What is the smallest number you must add to 110 to make it divisible by 4? (a) 3 (b) 1 (c) 2
8. 8.Which of these numbers is divisible by 3? (a) 1502 (b) 1499 (c) 1505 (d) 1500
9. 9.What is the remainder when 101 is divided by 5? (a) 0 (b) 2 (c) 1 (d) 4
10. 10.What is the smallest number you must add to 85 to make it divisible by 3? (a) 2 (b) 1
11. 11.What is the units digit of 3^28? (a) 7 (b) 1 (c) 9 (d) 3
12. 12.What is the units digit of 2^51? (a) 4 (b) 2 (c) 6 (d) 8
13. 13.What is the smallest number you must add to 471 to make it divisible by 9? (a) 5 (b) 3 (c) 7 (d) 6
14. 14.What is the remainder when 223 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
15. 15.Which of these numbers is divisible by 4? (a) 459 (b) 454 (c) 456 (d) 461
16. 16.What is the remainder when 182 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
17. 17.Which of these numbers is divisible by 11? (a) 1683 (b) 1703 (c) 1679 (d) 1690
18. 18.What is the smallest number you must add to 124 to make it divisible by 3? (a) 2 (b) 1
19. 19.What is the smallest number you must add to 235 to make it divisible by 8? (a) 4 (b) 6 (c) 3 (d) 5
20. 20.Which of these numbers is divisible by 6? (a) 3218 (b) 3223 (c) 3231 (d) 3222
21. 21.What is the smallest number you must add to 288 to make it divisible by 7? (a) 6 (b) 1 (c) 4 (d) 5
22. 22.Which of these numbers is divisible by 3? (a) 535 (b) 532 (c) 529 (d) 531
23. 23.What is the smallest number you must add to 192 to make it divisible by 9? (a) 3 (b) 6 (c) 5 (d) 7
24. 24.Which of these numbers is divisible by 8? (a) 1054 (b) 1056 (c) 1067 (d) 1059
25. 25.Which of these numbers is divisible by 3? (a) 2291 (b) 2289 (c) 2287 (d) 2294
26. 26.What is the units digit of 3^19? (a) 9 (b) 7 (c) 1 (d) 3
27. 27.What is the remainder when 505 is divided by 8? (a) 1 (b) 7 (c) 0 (d) 2
28. 28.What is the units digit of 4^28? (a) 4 (b) 7 (c) 5 (d) 6
29. 29.Which of these numbers is divisible by 6? (a) 1436 (b) 1432 (c) 1428 (d) 1425
30. 30.What is the units digit of 2^55? (a) 8 (b) 6 (c) 4 (d) 2
31. 31.Which of these numbers is divisible by 3? (a) 1682 (b) 1680 (c) 1678 (d) 1684
32. 32.Which of these numbers is divisible by 4? (a) 1569 (b) 1565 (c) 1568 (d) 1573
33. 33.What is the remainder when 435 is divided by 10? (a) 6 (b) 4 (c) 7 (d) 5
34. 34.Which of these numbers is divisible by 8? (a) 602 (b) 594 (c) 600 (d) 609
35. 35.What is the smallest number you must add to 106 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
36. 36.What is the units digit of 9^23? (a) 9 (b) 1 (c) 8 (d) 7
37. 37.What is the units digit of 2^28? (a) 4 (b) 2 (c) 8 (d) 6
38. 38.What is the smallest number you must add to 70 to make it divisible by 3? (a) 1 (b) 2
39. 39.What is the smallest number you must add to 158 to make it divisible by 8? (a) 2 (b) 6 (c) 3 (d) 1
40. 40.What is the units digit of 9^54? (a) 0 (b) 2 (c) 9 (d) 1

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