Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 685 · 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 9^22? (a) 1 (b) 2 (c) 9 (d) 0
2. 2.What is the units digit of 9^26? (a) 0 (b) 2 (c) 9 (d) 1
3. 3.Which of these numbers is divisible by 11? (a) 1464 (b) 1463 (c) 1480 (d) 1454
4. 4.What is the remainder when 197 is divided by 10? (a) 8 (b) 3 (c) 6 (d) 7
5. 5.What is the remainder when 239 is divided by 6? (a) 4 (b) 3 (c) 5 (d) 1
6. 6.What is the remainder when 580 is divided by 9? (a) 5 (b) 3 (c) 6 (d) 4
7. 7.What is the remainder when 288 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
8. 8.What is the remainder when 161 is divided by 11? (a) 6 (b) 7 (c) 8 (d) 4
9. 9.What is the units digit of 8^13? (a) 6 (b) 8 (c) 2 (d) 4
10. 10.What is the units digit of 2^15? (a) 2 (b) 8 (c) 4 (d) 6
11. 11.What is the remainder when 56 is divided by 5? (a) 4 (b) 0 (c) 1 (d) 2
12. 12.What is the smallest number you must add to 614 to make it divisible by 9? (a) 2 (b) 6 (c) 7 (d) 8
13. 13.What is the units digit of 4^31? (a) 6 (b) 4 (c) 5 (d) 3
14. 14.Which of these numbers is divisible by 9? (a) 6426 (b) 6429 (c) 6438 (d) 6420
15. 15.What is the remainder when 143 is divided by 12? (a) 9 (b) 11 (c) 1 (d) 10
16. 16.What is the units digit of 8^23? (a) 8 (b) 6 (c) 2 (d) 4
17. 17.What is the remainder when 221 is divided by 3? (a) 0 (b) 2 (c) 1
18. 18.What is the units digit of 3^11? (a) 3 (b) 9 (c) 1 (d) 7
19. 19.What is the units digit of 8^57? (a) 2 (b) 6 (c) 4 (d) 8
20. 20.What is the units digit of 9^35? (a) 9 (b) 8 (c) 7 (d) 1
21. 21.What is the units digit of 4^24? (a) 7 (b) 6 (c) 4 (d) 5
22. 22.What is the remainder when 289 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
23. 23.What is the smallest number you must add to 702 to make it divisible by 11? (a) 1 (b) 2 (c) 3 (d) 9
24. 24.What is the units digit of 3^20? (a) 3 (b) 1 (c) 7 (d) 9
25. 25.What is the smallest number you must add to 596 to make it divisible by 12? (a) 8 (b) 5 (c) 3 (d) 4
26. 26.What is the remainder when 610 is divided by 12? (a) 10 (b) 11 (c) 2 (d) 9
27. 27.What is the units digit of 3^28? (a) 1 (b) 3 (c) 9 (d) 7
28. 28.What is the smallest number you must add to 43 to make it divisible by 3? (a) 2 (b) 1
29. 29.What is the smallest number you must add to 64 to make it divisible by 3? (a) 1 (b) 2
30. 30.Which of these numbers is divisible by 3? (a) 1099 (b) 1098 (c) 1103 (d) 1097
31. 31.Which of these numbers is divisible by 9? (a) 6567 (b) 6552 (c) 6558 (d) 6550
32. 32.Which of these numbers is divisible by 9? (a) 2988 (b) 2996 (c) 3003 (d) 2981
33. 33.What is the units digit of 3^37? (a) 3 (b) 1 (c) 7 (d) 9
34. 34.What is the units digit of 3^45? (a) 7 (b) 1 (c) 9 (d) 3
35. 35.What is the remainder when 52 is divided by 3? (a) 2 (b) 0 (c) 1
36. 36.What is the smallest number you must add to 162 to make it divisible by 4? (a) 3 (b) 2 (c) 1
37. 37.What is the remainder when 677 is divided by 10? (a) 3 (b) 8 (c) 6 (d) 7
38. 38.What is the remainder when 51 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
39. 39.What is the units digit of 3^60? (a) 9 (b) 1 (c) 3 (d) 7
40. 40.Which of these numbers is divisible by 8? (a) 1642 (b) 1661 (c) 1652 (d) 1648

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