Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 713 · 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 units digit of 8^49? (a) 4 (b) 8 (c) 2 (d) 6
2. 2.What is the remainder when 189 is divided by 8? (a) 3 (b) 5 (c) 4 (d) 6
3. 3.What is the remainder when 245 is divided by 8? (a) 6 (b) 4 (c) 5 (d) 3
4. 4.What is the remainder when 109 is divided by 3? (a) 2 (b) 1 (c) 0
5. 5.What is the units digit of 8^53? (a) 8 (b) 6 (c) 2 (d) 4
6. 6.What is the remainder when 384 is divided by 9? (a) 5 (b) 7 (c) 6 (d) 3
7. 7.What is the units digit of 9^11? (a) 9 (b) 7 (c) 1 (d) 8
8. 8.What is the remainder when 230 is divided by 3? (a) 2 (b) 0 (c) 1
9. 9.Which of these numbers is divisible by 6? (a) 779 (b) 774 (c) 770 (d) 784
10. 10.What is the units digit of 7^53? (a) 7 (b) 9 (c) 3 (d) 1
11. 11.What is the remainder when 178 is divided by 3? (a) 2 (b) 1 (c) 0
12. 12.What is the smallest number you must add to 618 to make it divisible by 8? (a) 5 (b) 7 (c) 6 (d) 2
13. 13.What is the remainder when 47 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
14. 14.What is the units digit of 7^33? (a) 3 (b) 7 (c) 1 (d) 9
15. 15.Which of these numbers is divisible by 4? (a) 908 (b) 909 (c) 905 (d) 913
16. 16.What is the remainder when 151 is divided by 7? (a) 5 (b) 6 (c) 4 (d) 3
17. 17.Which of these numbers is divisible by 3? (a) 823 (b) 822 (c) 820 (d) 826
18. 18.What is the smallest number you must add to 685 to make it divisible by 11? (a) 7 (b) 3 (c) 8 (d) 9
19. 19.What is the units digit of 8^51? (a) 6 (b) 2 (c) 8 (d) 4
20. 20.Which of these numbers is divisible by 9? (a) 1136 (b) 1134 (c) 1146 (d) 1133
21. 21.What is the remainder when 201 is divided by 5? (a) 0 (b) 1 (c) 4 (d) 2
22. 22.What is the remainder when 61 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
23. 23.Which of these numbers is divisible by 8? (a) 6960 (b) 6969 (c) 6967 (d) 6958
24. 24.Which of these numbers is divisible by 3? (a) 1769 (b) 1772 (c) 1767 (d) 1766
25. 25.What is the smallest number you must add to 105 to make it divisible by 9? (a) 2 (b) 3 (c) 4 (d) 6
26. 26.What is the smallest number you must add to 314 to make it divisible by 5? (a) 2 (b) 4 (c) 1 (d) 3
27. 27.What is the remainder when 173 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
28. 28.Which of these numbers is divisible by 11? (a) 7706 (b) 7715 (c) 7700 (d) 7699
29. 29.What is the smallest number you must add to 86 to make it divisible by 4? (a) 2 (b) 1 (c) 3
30. 30.What is the smallest number you must add to 400 to make it divisible by 6? (a) 2 (b) 4 (c) 3 (d) 1
31. 31.What is the smallest number you must add to 327 to make it divisible by 11? (a) 8 (b) 3 (c) 4 (d) 2
32. 32.What is the smallest number you must add to 631 to make it divisible by 11? (a) 8 (b) 7 (c) 6 (d) 4
33. 33.Which of these numbers is divisible by 8? (a) 1967 (b) 1958 (c) 1951 (d) 1952
34. 34.What is the remainder when 411 is divided by 11? (a) 7 (b) 5 (c) 4 (d) 3
35. 35.What is the units digit of 8^36? (a) 6 (b) 8 (c) 4 (d) 2
36. 36.What is the remainder when 337 is divided by 9? (a) 5 (b) 3 (c) 6 (d) 4
37. 37.Which of these numbers is divisible by 8? (a) 2025 (b) 2021 (c) 2024 (d) 2035
38. 38.Which of these numbers is divisible by 9? (a) 2692 (b) 2703 (c) 2685 (d) 2691
39. 39.Which of these numbers is divisible by 6? (a) 982 (b) 978 (c) 987 (d) 975
40. 40.What is the units digit of 3^39? (a) 7 (b) 3 (c) 9 (d) 1

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