Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 981 · 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 8^32? (a) 2 (b) 4 (c) 6 (d) 8
2. 2.What is the smallest number you must add to 184 to make it divisible by 3? (a) 1 (b) 2
3. 3.What is the remainder when 379 is divided by 11? (a) 7 (b) 4 (c) 5 (d) 6
4. 4.What is the units digit of 9^20? (a) 1 (b) 0 (c) 2 (d) 9
5. 5.What is the units digit of 3^12? (a) 9 (b) 7 (c) 3 (d) 1
6. 6.Which of these numbers is divisible by 3? (a) 961 (b) 958 (c) 955 (d) 957
7. 7.What is the smallest number you must add to 430 to make it divisible by 8? (a) 6 (b) 3 (c) 2 (d) 1
8. 8.Which of these numbers is divisible by 6? (a) 4035 (b) 4038 (c) 4042 (d) 4048
9. 9.What is the units digit of 9^31? (a) 8 (b) 9 (c) 7 (d) 1
10. 10.Which of these numbers is divisible by 4? (a) 1795 (b) 1785 (c) 1788 (d) 1790
11. 11.What is the smallest number you must add to 67 to make it divisible by 3? (a) 1 (b) 2
12. 12.What is the units digit of 8^49? (a) 6 (b) 4 (c) 2 (d) 8
13. 13.What is the remainder when 274 is divided by 8? (a) 6 (b) 2 (c) 3 (d) 1
14. 14.Which of these numbers is divisible by 6? (a) 4017 (b) 4008 (c) 4005 (d) 4012
15. 15.What is the remainder when 794 is divided by 10? (a) 3 (b) 4 (c) 6 (d) 5
16. 16.Which of these numbers is divisible by 6? (a) 3772 (b) 3764 (c) 3762 (d) 3757
17. 17.What is the remainder when 536 is divided by 10? (a) 4 (b) 5 (c) 6 (d) 7
18. 18.What is the remainder when 271 is divided by 5? (a) 1 (b) 2 (c) 0 (d) 4
19. 19.What is the remainder when 317 is divided by 11? (a) 10 (b) 2 (c) 8 (d) 9
20. 20.Which of these numbers is divisible by 6? (a) 1645 (b) 1644 (c) 1641 (d) 1654
21. 21.Which of these numbers is divisible by 4? (a) 1605 (b) 1603 (c) 1599 (d) 1600
22. 22.What is the smallest number you must add to 265 to make it divisible by 6? (a) 1 (b) 5 (c) 3 (d) 4
23. 23.Which of these numbers is divisible by 3? (a) 657 (b) 655 (c) 659 (d) 662
24. 24.What is the units digit of 2^39? (a) 4 (b) 8 (c) 2 (d) 6
25. 25.What is the remainder when 74 is divided by 3? (a) 2 (b) 0 (c) 1
26. 26.What is the units digit of 2^52? (a) 4 (b) 8 (c) 2 (d) 6
27. 27.What is the smallest number you must add to 835 to make it divisible by 11? (a) 10 (b) 3 (c) 2 (d) 1
28. 28.What is the remainder when 178 is divided by 12? (a) 11 (b) 2 (c) 10 (d) 9
29. 29.Which of these numbers is divisible by 6? (a) 4036 (b) 4031 (c) 4026 (d) 4024
30. 30.Which of these numbers is divisible by 3? (a) 1552 (b) 1548 (c) 1550 (d) 1546
31. 31.What is the remainder when 308 is divided by 12? (a) 7 (b) 4 (c) 9 (d) 8
32. 32.What is the smallest number you must add to 116 to make it divisible by 7? (a) 4 (b) 5 (c) 2 (d) 3
33. 33.What is the units digit of 4^16? (a) 5 (b) 7 (c) 4 (d) 6
34. 34.What is the units digit of 2^23? (a) 4 (b) 6 (c) 2 (d) 8
35. 35.What is the units digit of 9^34? (a) 0 (b) 1 (c) 2 (d) 9
36. 36.What is the remainder when 110 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
37. 37.What is the smallest number you must add to 165 to make it divisible by 8? (a) 3 (b) 4 (c) 2 (d) 5
38. 38.What is the units digit of 7^42? (a) 1 (b) 3 (c) 7 (d) 9
39. 39.What is the units digit of 3^55? (a) 7 (b) 1 (c) 9 (d) 3
40. 40.What is the units digit of 2^22? (a) 6 (b) 2 (c) 8 (d) 4

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