Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 103 · 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 remainder when 379 is divided by 10? (a) 8 (b) 7 (c) 1 (d) 9
2. 2.What is the units digit of 4^35? (a) 3 (b) 4 (c) 6 (d) 5
3. 3.What is the smallest number you must add to 141 to make it divisible by 7? (a) 6 (b) 1 (c) 5 (d) 4
4. 4.Which of these numbers is divisible by 8? (a) 3996 (b) 3988 (c) 3984 (d) 3983
5. 5.What is the units digit of 4^36? (a) 4 (b) 7 (c) 5 (d) 6
6. 6.What is the units digit of 8^15? (a) 8 (b) 6 (c) 2 (d) 4
7. 7.What is the smallest number you must add to 55 to make it divisible by 3? (a) 1 (b) 2
8. 8.What is the smallest number you must add to 49 to make it divisible by 3? (a) 1 (b) 2
9. 9.What is the units digit of 3^34? (a) 7 (b) 3 (c) 1 (d) 9
10. 10.What is the smallest number you must add to 117 to make it divisible by 4? (a) 1 (b) 2 (c) 3
11. 11.What is the units digit of 8^24? (a) 6 (b) 8 (c) 2 (d) 4
12. 12.What is the smallest number you must add to 482 to make it divisible by 8? (a) 7 (b) 2 (c) 5 (d) 6
13. 13.What is the remainder when 361 is divided by 11? (a) 10 (b) 2 (c) 9 (d) 8
14. 14.What is the remainder when 143 is divided by 7? (a) 3 (b) 4 (c) 5 (d) 2
15. 15.What is the remainder when 303 is divided by 11? (a) 7 (b) 6 (c) 5 (d) 8
16. 16.What is the remainder when 61 is divided by 3? (a) 0 (b) 1 (c) 2
17. 17.Which of these numbers is divisible by 4? (a) 1295 (b) 1291 (c) 1297 (d) 1292
18. 18.What is the smallest number you must add to 761 to make it divisible by 10? (a) 1 (b) 7 (c) 9 (d) 8
19. 19.Which of these numbers is divisible by 8? (a) 6410 (b) 6418 (c) 6408 (d) 6403
20. 20.Which of these numbers is divisible by 9? (a) 6633 (b) 6650 (c) 6632 (d) 6634
21. 21.Which of these numbers is divisible by 3? (a) 740 (b) 738 (c) 736 (d) 743
22. 22.What is the units digit of 2^46? (a) 4 (b) 2 (c) 8 (d) 6
23. 23.What is the units digit of 3^31? (a) 1 (b) 7 (c) 3 (d) 9
24. 24.What is the remainder when 42 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
25. 25.What is the units digit of 4^24? (a) 4 (b) 6 (c) 7 (d) 5
26. 26.What is the units digit of 8^26? (a) 8 (b) 6 (c) 4 (d) 2
27. 27.What is the remainder when 118 is divided by 3? (a) 2 (b) 0 (c) 1
28. 28.Which of these numbers is divisible by 4? (a) 3114 (b) 3117 (c) 3111 (d) 3112
29. 29.What is the remainder when 119 is divided by 3? (a) 1 (b) 0 (c) 2
30. 30.What is the units digit of 8^56? (a) 8 (b) 2 (c) 4 (d) 6
31. 31.What is the smallest number you must add to 467 to make it divisible by 12? (a) 2 (b) 11 (c) 1 (d) 3
32. 32.What is the units digit of 7^46? (a) 1 (b) 3 (c) 7 (d) 9
33. 33.Which of these numbers is divisible by 6? (a) 385 (b) 378 (c) 374 (d) 383
34. 34.What is the remainder when 391 is divided by 6? (a) 2 (b) 5 (c) 1 (d) 0
35. 35.What is the remainder when 613 is divided by 10? (a) 2 (b) 3 (c) 4 (d) 7
36. 36.What is the remainder when 310 is divided by 11? (a) 1 (b) 9 (c) 3 (d) 2
37. 37.What is the units digit of 9^58? (a) 0 (b) 9 (c) 2 (d) 1
38. 38.Which of these numbers is divisible by 9? (a) 4177 (b) 4196 (c) 4191 (d) 4185
39. 39.What is the units digit of 3^60? (a) 9 (b) 7 (c) 1 (d) 3
40. 40.What is the smallest number you must add to 766 to make it divisible by 11? (a) 5 (b) 3 (c) 7 (d) 4

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