Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 47 · 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 3? (a) 635 (b) 631 (c) 628 (d) 630
2. 2.Which of these numbers is divisible by 11? (a) 1894 (b) 1879 (c) 1887 (d) 1881
3. 3.What is the remainder when 143 is divided by 3? (a) 2 (b) 1 (c) 0
4. 4.What is the remainder when 242 is divided by 6? (a) 1 (b) 3 (c) 4 (d) 2
5. 5.What is the smallest number you must add to 142 to make it divisible by 6? (a) 2 (b) 1 (c) 3 (d) 4
6. 6.What is the units digit of 4^40? (a) 5 (b) 6 (c) 4 (d) 7
7. 7.What is the remainder when 215 is divided by 9? (a) 6 (b) 7 (c) 8 (d) 1
8. 8.What is the units digit of 2^35? (a) 2 (b) 4 (c) 8 (d) 6
9. 9.What is the remainder when 524 is divided by 12? (a) 4 (b) 7 (c) 9 (d) 8
10. 10.What is the smallest number you must add to 93 to make it divisible by 5? (a) 4 (b) 3 (c) 1 (d) 2
11. 11.What is the units digit of 7^25? (a) 9 (b) 3 (c) 7 (d) 1
12. 12.What is the remainder when 151 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
13. 13.What is the units digit of 9^15? (a) 1 (b) 9 (c) 7 (d) 8
14. 14.Which of these numbers is divisible by 4? (a) 771 (b) 765 (c) 763 (d) 764
15. 15.What is the smallest number you must add to 432 to make it divisible by 7? (a) 5 (b) 1 (c) 2 (d) 3
16. 16.What is the smallest number you must add to 326 to make it divisible by 11? (a) 3 (b) 5 (c) 4 (d) 7
17. 17.What is the remainder when 99 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
18. 18.Which of these numbers is divisible by 9? (a) 2527 (b) 2534 (c) 2520 (d) 2518
19. 19.What is the smallest number you must add to 134 to make it divisible by 8? (a) 1 (b) 3 (c) 6 (d) 2
20. 20.Which of these numbers is divisible by 3? (a) 1916 (b) 1917 (c) 1919 (d) 1922
21. 21.Which of these numbers is divisible by 11? (a) 1595 (b) 1605 (c) 1590 (d) 1607
22. 22.What is the remainder when 212 is divided by 3? (a) 0 (b) 2 (c) 1
23. 23.Which of these numbers is divisible by 3? (a) 989 (b) 987 (c) 985 (d) 991
24. 24.What is the units digit of 4^18? (a) 5 (b) 7 (c) 4 (d) 6
25. 25.What is the remainder when 441 is divided by 6? (a) 2 (b) 4 (c) 5 (d) 3
26. 26.What is the remainder when 685 is divided by 11? (a) 2 (b) 4 (c) 8 (d) 3
27. 27.What is the units digit of 3^53? (a) 9 (b) 3 (c) 7 (d) 1
28. 28.Which of these numbers is divisible by 9? (a) 2124 (b) 2134 (c) 2122 (d) 2125
29. 29.What is the units digit of 8^15? (a) 2 (b) 6 (c) 8 (d) 4
30. 30.Which of these numbers is divisible by 6? (a) 2922 (b) 2926 (c) 2929 (d) 2920
31. 31.What is the smallest number you must add to 261 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
32. 32.Which of these numbers is divisible by 9? (a) 6282 (b) 6284 (c) 6297 (d) 6281
33. 33.Which of these numbers is divisible by 9? (a) 5013 (b) 5021 (c) 5005 (d) 5026
34. 34.What is the remainder when 397 is divided by 7? (a) 5 (b) 2 (c) 4 (d) 6
35. 35.Which of these numbers is divisible by 8? (a) 4398 (b) 4391 (c) 4404 (d) 4392
36. 36.What is the smallest number you must add to 219 to make it divisible by 10? (a) 9 (b) 1 (c) 2 (d) 3
37. 37.What is the remainder when 67 is divided by 3? (a) 2 (b) 0 (c) 1
38. 38.What is the units digit of 7^16? (a) 9 (b) 1 (c) 3 (d) 7
39. 39.Which of these numbers is divisible by 6? (a) 1694 (b) 1700 (c) 1689 (d) 1692
40. 40.What is the units digit of 3^37? (a) 1 (b) 9 (c) 3 (d) 7

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