Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 691 · 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 7^40? (a) 7 (b) 3 (c) 9 (d) 1
2. 2.What is the remainder when 177 is divided by 8? (a) 2 (b) 1 (c) 0 (d) 7
3. 3.What is the units digit of 7^24? (a) 3 (b) 7 (c) 1 (d) 9
4. 4.What is the smallest number you must add to 342 to make it divisible by 7? (a) 3 (b) 1 (c) 6 (d) 2
5. 5.What is the smallest number you must add to 116 to make it divisible by 3? (a) 2 (b) 1
6. 6.What is the units digit of 9^46? (a) 1 (b) 0 (c) 2 (d) 9
7. 7.What is the remainder when 296 is divided by 7? (a) 1 (b) 2 (c) 3 (d) 5
8. 8.Which of these numbers is divisible by 3? (a) 481 (b) 484 (c) 487 (d) 483
9. 9.Which of these numbers is divisible by 4? (a) 1473 (b) 1467 (c) 1471 (d) 1468
10. 10.Which of these numbers is divisible by 3? (a) 1132 (b) 1139 (c) 1135 (d) 1134
11. 11.What is the units digit of 8^58? (a) 4 (b) 2 (c) 8 (d) 6
12. 12.What is the units digit of 3^10? (a) 7 (b) 1 (c) 3 (d) 9
13. 13.What is the smallest number you must add to 255 to make it divisible by 12? (a) 9 (b) 8 (c) 3 (d) 10
14. 14.What is the remainder when 208 is divided by 7? (a) 5 (b) 2 (c) 6 (d) 4
15. 15.What is the units digit of 7^41? (a) 9 (b) 3 (c) 1 (d) 7
16. 16.What is the smallest number you must add to 203 to make it divisible by 3? (a) 2 (b) 1
17. 17.Which of these numbers is divisible by 9? (a) 7429 (b) 7447 (c) 7434 (d) 7438
18. 18.What is the units digit of 9^30? (a) 0 (b) 2 (c) 9 (d) 1
19. 19.Which of these numbers is divisible by 4? (a) 2947 (b) 2954 (c) 2948 (d) 2950
20. 20.Which of these numbers is divisible by 4? (a) 1537 (b) 1541 (c) 1545 (d) 1540
21. 21.What is the units digit of 3^13? (a) 7 (b) 9 (c) 1 (d) 3
22. 22.Which of these numbers is divisible by 6? (a) 1678 (b) 1680 (c) 1687 (d) 1684
23. 23.What is the units digit of 7^50? (a) 3 (b) 7 (c) 1 (d) 9
24. 24.What is the smallest number you must add to 201 to make it divisible by 8? (a) 6 (b) 5 (c) 1 (d) 7
25. 25.What is the smallest number you must add to 306 to make it divisible by 5? (a) 1 (b) 3 (c) 4 (d) 2
26. 26.What is the units digit of 8^16? (a) 6 (b) 8 (c) 2 (d) 4
27. 27.Which of these numbers is divisible by 8? (a) 4940 (b) 4944 (c) 4953 (d) 4948
28. 28.What is the remainder when 472 is divided by 10? (a) 3 (b) 1 (c) 2 (d) 8
29. 29.Which of these numbers is divisible by 11? (a) 574 (b) 563 (c) 572 (d) 592
30. 30.What is the remainder when 248 is divided by 6? (a) 2 (b) 1 (c) 4 (d) 3
31. 31.What is the smallest number you must add to 156 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
32. 32.What is the smallest number you must add to 97 to make it divisible by 7? (a) 2 (b) 1 (c) 3 (d) 6
33. 33.What is the smallest number you must add to 136 to make it divisible by 11? (a) 8 (b) 4 (c) 6 (d) 7
34. 34.What is the units digit of 7^35? (a) 3 (b) 7 (c) 9 (d) 1
35. 35.What is the smallest number you must add to 105 to make it divisible by 8? (a) 1 (b) 7 (c) 6 (d) 5
36. 36.What is the remainder when 199 is divided by 3? (a) 0 (b) 1 (c) 2
37. 37.What is the smallest number you must add to 164 to make it divisible by 3? (a) 2 (b) 1
38. 38.What is the remainder when 506 is divided by 12? (a) 1 (b) 10 (c) 2 (d) 3
39. 39.What is the remainder when 951 is divided by 12? (a) 4 (b) 2 (c) 3 (d) 9
40. 40.Which of these numbers is divisible by 4? (a) 247 (b) 243 (c) 249 (d) 244

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