Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 591 · 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 86 is divided by 6? (a) 4 (b) 3 (c) 2 (d) 1
2. 2.What is the units digit of 2^53? (a) 6 (b) 8 (c) 4 (d) 2
3. 3.What is the remainder when 108 is divided by 8? (a) 4 (b) 3 (c) 6 (d) 5
4. 4.Which of these numbers is divisible by 8? (a) 6851 (b) 6840 (c) 6839 (d) 6847
5. 5.What is the units digit of 9^58? (a) 0 (b) 1 (c) 2 (d) 9
6. 6.What is the remainder when 173 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
7. 7.Which of these numbers is divisible by 11? (a) 8712 (b) 8718 (c) 8732 (d) 8706
8. 8.What is the smallest number you must add to 173 to make it divisible by 4? (a) 2 (b) 3 (c) 1
9. 9.Which of these numbers is divisible by 3? (a) 1685 (b) 1687 (c) 1681 (d) 1683
10. 10.What is the remainder when 116 is divided by 6? (a) 4 (b) 1 (c) 2 (d) 3
11. 11.Which of these numbers is divisible by 6? (a) 1900 (b) 1903 (c) 1902 (d) 1909
12. 12.What is the remainder when 255 is divided by 9? (a) 3 (b) 6 (c) 2 (d) 4
13. 13.What is the smallest number you must add to 69 to make it divisible by 6? (a) 2 (b) 3 (c) 5 (d) 4
14. 14.What is the units digit of 9^18? (a) 0 (b) 2 (c) 9 (d) 1
15. 15.Which of these numbers is divisible by 3? (a) 2331 (b) 2330 (c) 2333 (d) 2335
16. 16.What is the remainder when 259 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
17. 17.What is the remainder when 723 is divided by 11? (a) 8 (b) 7 (c) 3 (d) 9
18. 18.What is the remainder when 641 is divided by 11? (a) 4 (b) 2 (c) 3 (d) 8
19. 19.What is the remainder when 449 is divided by 8? (a) 7 (b) 2 (c) 0 (d) 1
20. 20.Which of these numbers is divisible by 3? (a) 187 (b) 181 (c) 185 (d) 183
21. 21.Which of these numbers is divisible by 8? (a) 1943 (b) 1934 (c) 1936 (d) 1946
22. 22.Which of these numbers is divisible by 9? (a) 1529 (b) 1519 (c) 1512 (d) 1511
23. 23.What is the units digit of 9^59? (a) 1 (b) 7 (c) 8 (d) 9
24. 24.What is the smallest number you must add to 189 to make it divisible by 8? (a) 4 (b) 5 (c) 3 (d) 2
25. 25.What is the remainder when 59 is divided by 3? (a) 0 (b) 1 (c) 2
26. 26.What is the units digit of 3^24? (a) 7 (b) 1 (c) 3 (d) 9
27. 27.What is the smallest number you must add to 202 to make it divisible by 4? (a) 2 (b) 1 (c) 3
28. 28.What is the smallest number you must add to 445 to make it divisible by 10? (a) 4 (b) 5 (c) 7 (d) 6
29. 29.What is the smallest number you must add to 121 to make it divisible by 4? (a) 2 (b) 1 (c) 3
30. 30.What is the remainder when 568 is divided by 10? (a) 8 (b) 7 (c) 2 (d) 9
31. 31.What is the units digit of 8^33? (a) 2 (b) 6 (c) 4 (d) 8
32. 32.Which of these numbers is divisible by 6? (a) 1015 (b) 1020 (c) 1027 (d) 1021
33. 33.What is the units digit of 3^49? (a) 9 (b) 7 (c) 1 (d) 3
34. 34.What is the remainder when 311 is divided by 9? (a) 7 (b) 4 (c) 6 (d) 5
35. 35.Which of these numbers is divisible by 6? (a) 3814 (b) 3819 (c) 3807 (d) 3810
36. 36.Which of these numbers is divisible by 4? (a) 904 (b) 910 (c) 906 (d) 902
37. 37.What is the units digit of 4^48? (a) 7 (b) 5 (c) 4 (d) 6
38. 38.What is the units digit of 4^50? (a) 5 (b) 6 (c) 7 (d) 4
39. 39.What is the remainder when 631 is divided by 10? (a) 1 (b) 0 (c) 9 (d) 2
40. 40.What is the smallest number you must add to 217 to make it divisible by 4? (a) 1 (b) 2 (c) 3

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