Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 248 · 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 4^17? (a) 5 (b) 6 (c) 3 (d) 4
2. 2.What is the units digit of 2^12? (a) 2 (b) 6 (c) 8 (d) 4
3. 3.What is the units digit of 7^21? (a) 9 (b) 7 (c) 1 (d) 3
4. 4.What is the units digit of 4^23? (a) 3 (b) 5 (c) 4 (d) 6
5. 5.What is the smallest number you must add to 213 to make it divisible by 10? (a) 3 (b) 8 (c) 7 (d) 6
6. 6.What is the smallest number you must add to 458 to make it divisible by 12? (a) 10 (b) 9 (c) 2 (d) 11
7. 7.What is the remainder when 399 is divided by 9? (a) 2 (b) 6 (c) 4 (d) 3
8. 8.What is the remainder when 195 is divided by 6? (a) 4 (b) 2 (c) 3 (d) 5
9. 9.What is the units digit of 2^13? (a) 4 (b) 6 (c) 8 (d) 2
10. 10.What is the remainder when 478 is divided by 7? (a) 2 (b) 3 (c) 1 (d) 5
11. 11.What is the remainder when 186 is divided by 9? (a) 6 (b) 7 (c) 5 (d) 3
12. 12.Which of these numbers is divisible by 3? (a) 824 (b) 819 (c) 818 (d) 821
13. 13.Which of these numbers is divisible by 6? (a) 4077 (b) 4084 (c) 4089 (d) 4080
14. 14.What is the remainder when 133 is divided by 9? (a) 6 (b) 2 (c) 7 (d) 8
15. 15.What is the smallest number you must add to 453 to make it divisible by 12? (a) 9 (b) 3 (c) 2 (d) 4
16. 16.What is the units digit of 9^27? (a) 7 (b) 9 (c) 8 (d) 1
17. 17.What is the remainder when 351 is divided by 7? (a) 1 (b) 2 (c) 0 (d) 6
18. 18.What is the remainder when 333 is divided by 10? (a) 7 (b) 4 (c) 2 (d) 3
19. 19.What is the units digit of 2^43? (a) 6 (b) 2 (c) 8 (d) 4
20. 20.What is the smallest number you must add to 288 to make it divisible by 11? (a) 2 (b) 8 (c) 10 (d) 9
21. 21.What is the remainder when 511 is divided by 10? (a) 1 (b) 0 (c) 2 (d) 9
22. 22.What is the smallest number you must add to 577 to make it divisible by 10? (a) 7 (b) 4 (c) 2 (d) 3
23. 23.Which of these numbers is divisible by 4? (a) 2491 (b) 2495 (c) 2497 (d) 2492
24. 24.What is the remainder when 714 is divided by 12? (a) 6 (b) 8 (c) 7 (d) 5
25. 25.What is the units digit of 3^15? (a) 1 (b) 3 (c) 9 (d) 7
26. 26.Which of these numbers is divisible by 11? (a) 3443 (b) 3448 (c) 3458 (d) 3439
27. 27.What is the remainder when 482 is divided by 9? (a) 6 (b) 7 (c) 4 (d) 5
28. 28.What is the remainder when 158 is divided by 7? (a) 5 (b) 6 (c) 4 (d) 3
29. 29.What is the smallest number you must add to 65 to make it divisible by 4? (a) 1 (b) 2 (c) 3
30. 30.What is the remainder when 152 is divided by 11? (a) 2 (b) 10 (c) 9 (d) 8
31. 31.Which of these numbers is divisible by 8? (a) 6756 (b) 6760 (c) 6772 (d) 6764
32. 32.Which of these numbers is divisible by 11? (a) 4315 (b) 4294 (c) 4301 (d) 4305
33. 33.What is the units digit of 8^23? (a) 8 (b) 2 (c) 4 (d) 6
34. 34.What is the remainder when 154 is divided by 3? (a) 0 (b) 2 (c) 1
35. 35.Which of these numbers is divisible by 3? (a) 2120 (b) 2123 (c) 2116 (d) 2118
36. 36.What is the smallest number you must add to 471 to make it divisible by 6? (a) 5 (b) 3 (c) 4 (d) 2
37. 37.What is the remainder when 661 is divided by 9? (a) 5 (b) 3 (c) 6 (d) 4
38. 38.What is the remainder when 430 is divided by 12? (a) 9 (b) 11 (c) 10 (d) 2
39. 39.What is the remainder when 439 is divided by 8? (a) 6 (b) 5 (c) 1 (d) 7
40. 40.What is the units digit of 3^29? (a) 1 (b) 9 (c) 3 (d) 7

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