Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 737 · 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 7^42? (a) 1 (b) 7 (c) 9 (d) 3
2. 2.What is the smallest number you must add to 219 to make it divisible by 4? (a) 3 (b) 1 (c) 2
3. 3.What is the smallest number you must add to 138 to make it divisible by 10? (a) 3 (b) 2 (c) 1 (d) 8
4. 4.Which of these numbers is divisible by 9? (a) 5231 (b) 5244 (c) 5229 (d) 5225
5. 5.Which of these numbers is divisible by 3? (a) 141 (b) 142 (c) 146 (d) 139
6. 6.What is the smallest number you must add to 554 to make it divisible by 11? (a) 6 (b) 4 (c) 7 (d) 8
7. 7.Which of these numbers is divisible by 3? (a) 1471 (b) 1475 (c) 1470 (d) 1469
8. 8.What is the remainder when 113 is divided by 9? (a) 5 (b) 6 (c) 7 (d) 4
9. 9.What is the units digit of 2^30? (a) 6 (b) 8 (c) 4 (d) 2
10. 10.What is the remainder when 494 is divided by 10? (a) 3 (b) 4 (c) 5 (d) 6
11. 11.What is the remainder when 147 is divided by 6? (a) 3 (b) 4 (c) 2 (d) 5
12. 12.What is the remainder when 198 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
13. 13.Which of these numbers is divisible by 6? (a) 1481 (b) 1470 (c) 1465 (d) 1475
14. 14.What is the smallest number you must add to 254 to make it divisible by 7? (a) 6 (b) 2 (c) 5 (d) 4
15. 15.What is the units digit of 2^15? (a) 2 (b) 8 (c) 4 (d) 6
16. 16.What is the remainder when 593 is divided by 9? (a) 6 (b) 1 (c) 8 (d) 7
17. 17.Which of these numbers is divisible by 9? (a) 8051 (b) 8060 (c) 8040 (d) 8046
18. 18.What is the units digit of 4^16? (a) 5 (b) 6 (c) 7 (d) 4
19. 19.Which of these numbers is divisible by 8? (a) 4288 (b) 4282 (c) 4290 (d) 4298
20. 20.What is the units digit of 8^23? (a) 2 (b) 6 (c) 8 (d) 4
21. 21.Which of these numbers is divisible by 9? (a) 4994 (b) 4977 (c) 4985 (d) 4972
22. 22.What is the remainder when 263 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
23. 23.What is the remainder when 429 is divided by 9? (a) 3 (b) 6 (c) 5 (d) 7
24. 24.What is the units digit of 4^25? (a) 4 (b) 5 (c) 3 (d) 6
25. 25.What is the smallest number you must add to 358 to make it divisible by 8? (a) 6 (b) 2 (c) 1 (d) 3
26. 26.What is the remainder when 77 is divided by 6? (a) 1 (b) 3 (c) 5 (d) 4
27. 27.Which of these numbers is divisible by 6? (a) 1517 (b) 1511 (c) 1502 (d) 1506
28. 28.What is the units digit of 2^27? (a) 8 (b) 2 (c) 6 (d) 4
29. 29.What is the remainder when 62 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0
30. 30.What is the remainder when 176 is divided by 7? (a) 0 (b) 6 (c) 2 (d) 1
31. 31.Which of these numbers is divisible by 4? (a) 425 (b) 424 (c) 431 (d) 422
32. 32.Which of these numbers is divisible by 11? (a) 7040 (b) 7030 (c) 7055 (d) 7047
33. 33.What is the remainder when 611 is divided by 10? (a) 2 (b) 0 (c) 9 (d) 1
34. 34.Which of these numbers is divisible by 8? (a) 6548 (b) 6536 (c) 6532 (d) 6537
35. 35.Which of these numbers is divisible by 6? (a) 569 (b) 558 (c) 553 (d) 562
36. 36.Which of these numbers is divisible by 8? (a) 3534 (b) 3546 (c) 3539 (d) 3536
37. 37.What is the remainder when 387 is divided by 6? (a) 3 (b) 2 (c) 5 (d) 4
38. 38.What is the units digit of 8^48? (a) 8 (b) 2 (c) 4 (d) 6
39. 39.What is the smallest number you must add to 257 to make it divisible by 8? (a) 7 (b) 5 (c) 6 (d) 1
40. 40.What is the smallest number you must add to 155 to make it divisible by 3? (a) 2 (b) 1

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