Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 950 · 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 7^39? (a) 1 (b) 9 (c) 3 (d) 7
2. 2.What is the units digit of 7^50? (a) 9 (b) 1 (c) 3 (d) 7
3. 3.What is the units digit of 9^21? (a) 8 (b) 7 (c) 1 (d) 9
4. 4.What is the remainder when 431 is divided by 10? (a) 2 (b) 0 (c) 1 (d) 9
5. 5.What is the smallest number you must add to 630 to make it divisible by 8? (a) 2 (b) 3 (c) 1 (d) 6
6. 6.What is the units digit of 2^28? (a) 4 (b) 8 (c) 6 (d) 2
7. 7.What is the smallest number you must add to 427 to make it divisible by 10? (a) 4 (b) 7 (c) 2 (d) 3
8. 8.What is the smallest number you must add to 83 to make it divisible by 6? (a) 3 (b) 2 (c) 1 (d) 5
9. 9.What is the units digit of 9^31? (a) 8 (b) 9 (c) 7 (d) 1
10. 10.What is the units digit of 7^20? (a) 3 (b) 7 (c) 9 (d) 1
11. 11.What is the units digit of 2^44? (a) 8 (b) 2 (c) 6 (d) 4
12. 12.What is the remainder when 166 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
13. 13.Which of these numbers is divisible by 3? (a) 435 (b) 436 (c) 434 (d) 440
14. 14.What is the smallest number you must add to 285 to make it divisible by 12? (a) 9 (b) 4 (c) 2 (d) 3
15. 15.What is the units digit of 3^40? (a) 1 (b) 9 (c) 3 (d) 7
16. 16.Which of these numbers is divisible by 6? (a) 648 (b) 651 (c) 643 (d) 659
17. 17.Which of these numbers is divisible by 11? (a) 6451 (b) 6429 (c) 6438 (d) 6435
18. 18.What is the smallest number you must add to 237 to make it divisible by 7? (a) 6 (b) 2 (c) 3 (d) 1
19. 19.What is the remainder when 642 is divided by 8? (a) 6 (b) 1 (c) 3 (d) 2
20. 20.Which of these numbers is divisible by 3? (a) 2520 (b) 2524 (c) 2522 (d) 2519
21. 21.What is the units digit of 9^16? (a) 9 (b) 1 (c) 0 (d) 2
22. 22.What is the smallest number you must add to 687 to make it divisible by 12? (a) 8 (b) 10 (c) 9 (d) 3
23. 23.Which of these numbers is divisible by 8? (a) 6040 (b) 6055 (c) 6045 (d) 6033
24. 24.What is the smallest number you must add to 120 to make it divisible by 7? (a) 1 (b) 5 (c) 4 (d) 6
25. 25.Which of these numbers is divisible by 3? (a) 1251 (b) 1250 (c) 1252 (d) 1256
26. 26.What is the smallest number you must add to 589 to make it divisible by 9? (a) 5 (b) 6 (c) 4 (d) 7
27. 27.What is the remainder when 59 is divided by 3? (a) 0 (b) 1 (c) 2
28. 28.What is the units digit of 3^60? (a) 9 (b) 3 (c) 1 (d) 7
29. 29.What is the units digit of 7^40? (a) 3 (b) 1 (c) 7 (d) 9
30. 30.What is the smallest number you must add to 218 to make it divisible by 10? (a) 2 (b) 3 (c) 8 (d) 1
31. 31.Which of these numbers is divisible by 4? (a) 3109 (b) 3108 (c) 3107 (d) 3113
32. 32.What is the smallest number you must add to 66 to make it divisible by 4? (a) 1 (b) 3 (c) 2
33. 33.Which of these numbers is divisible by 3? (a) 2455 (b) 2459 (c) 2454 (d) 2453
34. 34.What is the remainder when 140 is divided by 6? (a) 4 (b) 3 (c) 2 (d) 1
35. 35.What is the smallest number you must add to 195 to make it divisible by 11? (a) 2 (b) 4 (c) 8 (d) 3
36. 36.What is the remainder when 281 is divided by 4? (a) 2 (b) 3 (c) 0 (d) 1
37. 37.Which of these numbers is divisible by 11? (a) 4070 (b) 4074 (c) 4084 (d) 4068
38. 38.What is the smallest number you must add to 224 to make it divisible by 10? (a) 7 (b) 5 (c) 6 (d) 4
39. 39.What is the units digit of 7^29? (a) 9 (b) 1 (c) 7 (d) 3
40. 40.What is the remainder when 482 is divided by 8? (a) 6 (b) 3 (c) 1 (d) 2

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