Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 210 · 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 9? (a) 8089 (b) 8093 (c) 8091 (d) 8102
2. 2.What is the units digit of 2^19? (a) 8 (b) 6 (c) 4 (d) 2
3. 3.What is the smallest number you must add to 121 to make it divisible by 3? (a) 1 (b) 2
4. 4.What is the smallest number you must add to 389 to make it divisible by 8? (a) 5 (b) 3 (c) 4 (d) 2
5. 5.What is the units digit of 2^26? (a) 8 (b) 2 (c) 6 (d) 4
6. 6.What is the smallest number you must add to 151 to make it divisible by 3? (a) 1 (b) 2
7. 7.What is the remainder when 582 is divided by 9? (a) 3 (b) 5 (c) 6 (d) 7
8. 8.What is the smallest number you must add to 131 to make it divisible by 4? (a) 2 (b) 3 (c) 1
9. 9.What is the remainder when 360 is divided by 7? (a) 2 (b) 4 (c) 5 (d) 3
10. 10.Which of these numbers is divisible by 6? (a) 610 (b) 621 (c) 612 (d) 615
11. 11.What is the units digit of 7^46? (a) 7 (b) 9 (c) 1 (d) 3
12. 12.Which of these numbers is divisible by 4? (a) 3205 (b) 3197 (c) 3200 (d) 3202
13. 13.What is the units digit of 8^35? (a) 6 (b) 4 (c) 8 (d) 2
14. 14.What is the units digit of 4^25? (a) 3 (b) 4 (c) 5 (d) 6
15. 15.What is the units digit of 4^32? (a) 4 (b) 5 (c) 7 (d) 6
16. 16.Which of these numbers is divisible by 8? (a) 1168 (b) 1164 (c) 1172 (d) 1178
17. 17.Which of these numbers is divisible by 4? (a) 2600 (b) 2601 (c) 2607 (d) 2598
18. 18.Which of these numbers is divisible by 9? (a) 2886 (b) 2891 (c) 2901 (d) 2889
19. 19.What is the units digit of 8^45? (a) 2 (b) 6 (c) 8 (d) 4
20. 20.Which of these numbers is divisible by 4? (a) 3447 (b) 3451 (c) 3443 (d) 3444
21. 21.What is the remainder when 142 is divided by 12? (a) 2 (b) 9 (c) 10 (d) 11
22. 22.Which of these numbers is divisible by 3? (a) 1609 (b) 1604 (c) 1606 (d) 1605
23. 23.What is the units digit of 7^45? (a) 3 (b) 7 (c) 1 (d) 9
24. 24.What is the units digit of 2^47? (a) 4 (b) 6 (c) 2 (d) 8
25. 25.What is the units digit of 7^22? (a) 3 (b) 7 (c) 1 (d) 9
26. 26.What is the units digit of 7^34? (a) 3 (b) 1 (c) 7 (d) 9
27. 27.What is the units digit of 4^17? (a) 6 (b) 5 (c) 3 (d) 4
28. 28.Which of these numbers is divisible by 9? (a) 2158 (b) 2168 (c) 2160 (d) 2171
29. 29.What is the smallest number you must add to 167 to make it divisible by 11? (a) 2 (b) 8 (c) 9 (d) 10
30. 30.What is the remainder when 175 is divided by 3? (a) 1 (b) 2 (c) 0
31. 31.Which of these numbers is divisible by 3? (a) 197 (b) 194 (c) 199 (d) 195
32. 32.Which of these numbers is divisible by 6? (a) 3023 (b) 3032 (c) 3024 (d) 3028
33. 33.Which of these numbers is divisible by 3? (a) 910 (b) 908 (c) 905 (d) 906
34. 34.What is the smallest number you must add to 131 to make it divisible by 3? (a) 1 (b) 2
35. 35.What is the units digit of 9^32? (a) 0 (b) 9 (c) 2 (d) 1
36. 36.Which of these numbers is divisible by 11? (a) 3092 (b) 3087 (c) 3091 (d) 3112
37. 37.Which of these numbers is divisible by 4? (a) 831 (b) 828 (c) 834 (d) 826
38. 38.What is the units digit of 4^14? (a) 7 (b) 4 (c) 5 (d) 6
39. 39.Which of these numbers is divisible by 6? (a) 1177 (b) 1170 (c) 1168 (d) 1175
40. 40.Which of these numbers is divisible by 6? (a) 5232 (b) 5243 (c) 5233 (d) 5230

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