Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 220 · 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 4? (a) 184 (b) 190 (c) 187 (d) 181
2. 2.What is the units digit of 2^56? (a) 8 (b) 2 (c) 6 (d) 4
3. 3.What is the remainder when 318 is divided by 10? (a) 9 (b) 7 (c) 2 (d) 8
4. 4.What is the remainder when 265 is divided by 9? (a) 3 (b) 5 (c) 6 (d) 4
5. 5.What is the units digit of 4^54? (a) 6 (b) 4 (c) 5 (d) 7
6. 6.What is the units digit of 4^20? (a) 5 (b) 4 (c) 6 (d) 7
7. 7.Which of these numbers is divisible by 6? (a) 1583 (b) 1576 (c) 1572 (d) 1568
8. 8.What is the remainder when 54 is divided by 5? (a) 3 (b) 4 (c) 1 (d) 2
9. 9.What is the units digit of 8^19? (a) 6 (b) 4 (c) 8 (d) 2
10. 10.What is the remainder when 725 is divided by 9? (a) 5 (b) 7 (c) 4 (d) 6
11. 11.Which of these numbers is divisible by 6? (a) 984 (b) 988 (c) 981 (d) 995
12. 12.What is the units digit of 9^18? (a) 1 (b) 9 (c) 0 (d) 2
13. 13.What is the remainder when 265 is divided by 6? (a) 2 (b) 0 (c) 1 (d) 5
14. 14.What is the units digit of 9^16? (a) 2 (b) 1 (c) 9 (d) 0
15. 15.What is the remainder when 96 is divided by 9? (a) 7 (b) 5 (c) 3 (d) 6
16. 16.Which of these numbers is divisible by 3? (a) 2084 (b) 2079 (c) 2081 (d) 2077
17. 17.What is the units digit of 8^34? (a) 4 (b) 2 (c) 6 (d) 8
18. 18.What is the units digit of 9^49? (a) 1 (b) 7 (c) 8 (d) 9
19. 19.What is the units digit of 3^28? (a) 9 (b) 3 (c) 7 (d) 1
20. 20.What is the remainder when 231 is divided by 5? (a) 2 (b) 4 (c) 0 (d) 1
21. 21.What is the remainder when 261 is divided by 7? (a) 1 (b) 2 (c) 5 (d) 3
22. 22.What is the units digit of 9^33? (a) 1 (b) 7 (c) 8 (d) 9
23. 23.What is the remainder when 163 is divided by 9? (a) 2 (b) 0 (c) 1 (d) 8
24. 24.Which of these numbers is divisible by 11? (a) 7016 (b) 7025 (c) 7032 (d) 7018
25. 25.Which of these numbers is divisible by 3? (a) 1210 (b) 1207 (c) 1206 (d) 1204
26. 26.Which of these numbers is divisible by 6? (a) 3644 (b) 3642 (c) 3653 (d) 3639
27. 27.Which of these numbers is divisible by 11? (a) 970 (b) 953 (c) 960 (d) 957
28. 28.What is the remainder when 486 is divided by 12? (a) 5 (b) 6 (c) 8 (d) 7
29. 29.What is the units digit of 8^47? (a) 4 (b) 6 (c) 8 (d) 2
30. 30.What is the units digit of 3^52? (a) 9 (b) 7 (c) 3 (d) 1
31. 31.What is the units digit of 3^49? (a) 1 (b) 9 (c) 3 (d) 7
32. 32.What is the remainder when 239 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
33. 33.Which of these numbers is divisible by 3? (a) 197 (b) 194 (c) 192 (d) 190
34. 34.Which of these numbers is divisible by 3? (a) 1846 (b) 1849 (c) 1852 (d) 1848
35. 35.What is the remainder when 314 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
36. 36.What is the smallest number you must add to 235 to make it divisible by 3? (a) 1 (b) 2
37. 37.Which of these numbers is divisible by 3? (a) 563 (b) 559 (c) 556 (d) 558
38. 38.What is the units digit of 8^23? (a) 2 (b) 8 (c) 6 (d) 4
39. 39.What is the remainder when 589 is divided by 9? (a) 5 (b) 4 (c) 3 (d) 6
40. 40.What is the units digit of 8^12? (a) 6 (b) 2 (c) 4 (d) 8

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