Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 360 · 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 6? (a) 3363 (b) 3374 (c) 3366 (d) 3368
2. 2.What is the smallest number you must add to 340 to make it divisible by 9? (a) 3 (b) 2 (c) 7 (d) 1
3. 3.What is the smallest number you must add to 348 to make it divisible by 7? (a) 5 (b) 2 (c) 3 (d) 1
4. 4.Which of these numbers is divisible by 6? (a) 1141 (b) 1155 (c) 1148 (d) 1146
5. 5.What is the units digit of 8^12? (a) 6 (b) 4 (c) 2 (d) 8
6. 6.What is the units digit of 8^10? (a) 2 (b) 6 (c) 8 (d) 4
7. 7.What is the smallest number you must add to 248 to make it divisible by 6? (a) 5 (b) 2 (c) 4 (d) 3
8. 8.Which of these numbers is divisible by 9? (a) 2481 (b) 2484 (c) 2495 (d) 2487
9. 9.What is the smallest number you must add to 554 to make it divisible by 10? (a) 6 (b) 4 (c) 7 (d) 5
10. 10.What is the smallest number you must add to 483 to make it divisible by 8? (a) 5 (b) 3 (c) 6 (d) 4
11. 11.What is the smallest number you must add to 942 to make it divisible by 12? (a) 5 (b) 6 (c) 7 (d) 8
12. 12.Which of these numbers is divisible by 9? (a) 3151 (b) 3150 (c) 3166 (d) 3147
13. 13.What is the units digit of 9^15? (a) 9 (b) 7 (c) 8 (d) 1
14. 14.Which of these numbers is divisible by 8? (a) 4285 (b) 4279 (c) 4292 (d) 4280
15. 15.What is the units digit of 7^29? (a) 3 (b) 7 (c) 1 (d) 9
16. 16.What is the units digit of 3^13? (a) 7 (b) 3 (c) 1 (d) 9
17. 17.What is the units digit of 4^35? (a) 4 (b) 5 (c) 3 (d) 6
18. 18.Which of these numbers is divisible by 4? (a) 3485 (b) 3483 (c) 3480 (d) 3478
19. 19.What is the units digit of 2^31? (a) 6 (b) 8 (c) 2 (d) 4
20. 20.What is the remainder when 109 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
21. 21.What is the smallest number you must add to 239 to make it divisible by 8? (a) 2 (b) 3 (c) 7 (d) 1
22. 22.What is the smallest number you must add to 420 to make it divisible by 8? (a) 3 (b) 4 (c) 6 (d) 5
23. 23.What is the smallest number you must add to 258 to make it divisible by 4? (a) 3 (b) 1 (c) 2
24. 24.Which of these numbers is divisible by 9? (a) 6799 (b) 6815 (c) 6808 (d) 6804
25. 25.What is the units digit of 4^50? (a) 5 (b) 4 (c) 7 (d) 6
26. 26.What is the smallest number you must add to 259 to make it divisible by 11? (a) 7 (b) 6 (c) 5 (d) 4
27. 27.What is the units digit of 4^51? (a) 5 (b) 6 (c) 4 (d) 3
28. 28.What is the remainder when 369 is divided by 8? (a) 2 (b) 1 (c) 0 (d) 7
29. 29.Which of these numbers is divisible by 9? (a) 6195 (b) 6187 (c) 6183 (d) 6175
30. 30.What is the remainder when 558 is divided by 8? (a) 2 (b) 5 (c) 6 (d) 7
31. 31.Which of these numbers is divisible by 4? (a) 1098 (b) 1095 (c) 1091 (d) 1092
32. 32.Which of these numbers is divisible by 9? (a) 979 (b) 994 (c) 981 (d) 985
33. 33.Which of these numbers is divisible by 3? (a) 2149 (b) 2148 (c) 2152 (d) 2146
34. 34.What is the remainder when 112 is divided by 3? (a) 0 (b) 1 (c) 2
35. 35.Which of these numbers is divisible by 8? (a) 6136 (b) 6138 (c) 6149 (d) 6130
36. 36.What is the smallest number you must add to 146 to make it divisible by 4? (a) 1 (b) 2 (c) 3
37. 37.What is the units digit of 3^55? (a) 3 (b) 9 (c) 7 (d) 1
38. 38.What is the units digit of 3^34? (a) 9 (b) 7 (c) 3 (d) 1
39. 39.What is the remainder when 155 is divided by 8? (a) 5 (b) 2 (c) 4 (d) 3
40. 40.Which of these numbers is divisible by 11? (a) 2673 (b) 2687 (c) 2678 (d) 2672

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