Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 941 · 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 smallest number you must add to 146 to make it divisible by 3? (a) 1 (b) 2
2. 2.What is the units digit of 9^48? (a) 9 (b) 2 (c) 0 (d) 1
3. 3.What is the smallest number you must add to 602 to make it divisible by 11? (a) 3 (b) 8 (c) 2 (d) 4
4. 4.Which of these numbers is divisible by 4? (a) 3142 (b) 3139 (c) 3136 (d) 3134
5. 5.What is the units digit of 8^27? (a) 2 (b) 6 (c) 4 (d) 8
6. 6.Which of these numbers is divisible by 3? (a) 717 (b) 715 (c) 722 (d) 718
7. 7.What is the remainder when 221 is divided by 10? (a) 0 (b) 9 (c) 1 (d) 2
8. 8.What is the remainder when 92 is divided by 9? (a) 1 (b) 7 (c) 3 (d) 2
9. 9.What is the smallest number you must add to 314 to make it divisible by 8? (a) 6 (b) 5 (c) 2 (d) 7
10. 10.What is the units digit of 7^30? (a) 7 (b) 3 (c) 9 (d) 1
11. 11.What is the smallest number you must add to 486 to make it divisible by 12? (a) 6 (b) 5 (c) 8 (d) 7
12. 12.What is the remainder when 74 is divided by 5? (a) 2 (b) 4 (c) 3 (d) 1
13. 13.What is the units digit of 4^37? (a) 5 (b) 4 (c) 6 (d) 3
14. 14.Which of these numbers is divisible by 3? (a) 2122 (b) 2121 (c) 2126 (d) 2120
15. 15.What is the smallest number you must add to 447 to make it divisible by 6? (a) 4 (b) 3 (c) 2 (d) 5
16. 16.What is the remainder when 317 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
17. 17.Which of these numbers is divisible by 4? (a) 682 (b) 686 (c) 678 (d) 680
18. 18.Which of these numbers is divisible by 3? (a) 652 (b) 655 (c) 659 (d) 654
19. 19.What is the units digit of 9^47? (a) 1 (b) 8 (c) 7 (d) 9
20. 20.What is the units digit of 3^21? (a) 9 (b) 1 (c) 3 (d) 7
21. 21.Which of these numbers is divisible by 4? (a) 358 (b) 360 (c) 363 (d) 365
22. 22.What is the remainder when 166 is divided by 3? (a) 0 (b) 2 (c) 1
23. 23.What is the smallest number you must add to 142 to make it divisible by 3? (a) 1 (b) 2
24. 24.Which of these numbers is divisible by 4? (a) 1136 (b) 1134 (c) 1138 (d) 1142
25. 25.Which of these numbers is divisible by 9? (a) 1274 (b) 1266 (c) 1260 (d) 1253
26. 26.What is the remainder when 227 is divided by 3? (a) 2 (b) 0 (c) 1
27. 27.What is the units digit of 8^52? (a) 8 (b) 4 (c) 6 (d) 2
28. 28.What is the smallest number you must add to 134 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
29. 29.What is the smallest number you must add to 212 to make it divisible by 10? (a) 7 (b) 9 (c) 8 (d) 2
30. 30.What is the smallest number you must add to 391 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
31. 31.What is the smallest number you must add to 135 to make it divisible by 7? (a) 2 (b) 6 (c) 4 (d) 5
32. 32.Which of these numbers is divisible by 11? (a) 9636 (b) 9652 (c) 9627 (d) 9645
33. 33.What is the units digit of 8^49? (a) 6 (b) 8 (c) 4 (d) 2
34. 34.What is the units digit of 4^35? (a) 5 (b) 6 (c) 4 (d) 3
35. 35.What is the remainder when 376 is divided by 9? (a) 6 (b) 2 (c) 8 (d) 7
36. 36.What is the remainder when 370 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 2
37. 37.Which of these numbers is divisible by 9? (a) 3764 (b) 3777 (c) 3782 (d) 3771
38. 38.What is the units digit of 8^36? (a) 8 (b) 4 (c) 2 (d) 6
39. 39.What is the units digit of 4^33? (a) 3 (b) 5 (c) 6 (d) 4
40. 40.What is the units digit of 8^15? (a) 6 (b) 4 (c) 2 (d) 8

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