Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 138 · 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 677 is divided by 11? (a) 5 (b) 7 (c) 6 (d) 8
2. 2.Which of these numbers is divisible by 8? (a) 3365 (b) 3358 (c) 3360 (d) 3375
3. 3.What is the remainder when 318 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
4. 4.What is the units digit of 3^24? (a) 3 (b) 7 (c) 1 (d) 9
5. 5.What is the remainder when 712 is divided by 12? (a) 3 (b) 5 (c) 4 (d) 8
6. 6.What is the units digit of 2^59? (a) 6 (b) 8 (c) 4 (d) 2
7. 7.What is the remainder when 303 is divided by 6? (a) 4 (b) 3 (c) 2 (d) 5
8. 8.What is the units digit of 9^31? (a) 7 (b) 1 (c) 8 (d) 9
9. 9.What is the smallest number you must add to 638 to make it divisible by 10? (a) 3 (b) 8 (c) 1 (d) 2
10. 10.What is the smallest number you must add to 133 to make it divisible by 3? (a) 2 (b) 1
11. 11.What is the remainder when 133 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
12. 12.What is the smallest number you must add to 131 to make it divisible by 7? (a) 5 (b) 3 (c) 2 (d) 1
13. 13.What is the units digit of 7^57? (a) 9 (b) 1 (c) 7 (d) 3
14. 14.What is the units digit of 3^11? (a) 9 (b) 1 (c) 3 (d) 7
15. 15.Which of these numbers is divisible by 4? (a) 1603 (b) 1600 (c) 1606 (d) 1598
16. 16.What is the remainder when 169 is divided by 5? (a) 3 (b) 2 (c) 4 (d) 1
17. 17.Which of these numbers is divisible by 9? (a) 1302 (b) 1311 (c) 1321 (d) 1305
18. 18.What is the remainder when 577 is divided by 8? (a) 2 (b) 1 (c) 0 (d) 7
19. 19.What is the units digit of 7^45? (a) 1 (b) 7 (c) 9 (d) 3
20. 20.Which of these numbers is divisible by 9? (a) 5691 (b) 5682 (c) 5688 (d) 5703
21. 21.What is the remainder when 557 is divided by 8? (a) 4 (b) 3 (c) 5 (d) 6
22. 22.What is the units digit of 3^13? (a) 3 (b) 9 (c) 1 (d) 7
23. 23.Which of these numbers is divisible by 8? (a) 6441 (b) 6432 (c) 6428 (d) 6438
24. 24.Which of these numbers is divisible by 6? (a) 1637 (b) 1628 (c) 1626 (d) 1624
25. 25.What is the smallest number you must add to 152 to make it divisible by 3? (a) 1 (b) 2
26. 26.What is the remainder when 64 is divided by 6? (a) 5 (b) 3 (c) 2 (d) 4
27. 27.What is the units digit of 4^54? (a) 6 (b) 7 (c) 4 (d) 5
28. 28.What is the remainder when 267 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
29. 29.What is the smallest number you must add to 239 to make it divisible by 3? (a) 1 (b) 2
30. 30.Which of these numbers is divisible by 8? (a) 5669 (b) 5650 (c) 5658 (d) 5656
31. 31.What is the remainder when 255 is divided by 7? (a) 5 (b) 3 (c) 4 (d) 2
32. 32.Which of these numbers is divisible by 9? (a) 2975 (b) 2966 (c) 2986 (d) 2970
33. 33.What is the units digit of 8^52? (a) 2 (b) 6 (c) 8 (d) 4
34. 34.What is the units digit of 8^24? (a) 4 (b) 2 (c) 8 (d) 6
35. 35.What is the units digit of 3^14? (a) 9 (b) 7 (c) 1 (d) 3
36. 36.What is the smallest number you must add to 359 to make it divisible by 9? (a) 3 (b) 2 (c) 8 (d) 1
37. 37.What is the smallest number you must add to 154 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
38. 38.What is the smallest number you must add to 116 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
39. 39.What is the units digit of 8^15? (a) 8 (b) 6 (c) 2 (d) 4
40. 40.What is the smallest number you must add to 560 to make it divisible by 11? (a) 1 (b) 3 (c) 10 (d) 2

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