Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 990 · 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 3^18? (a) 7 (b) 3 (c) 1 (d) 9
2. 2.What is the units digit of 7^24? (a) 3 (b) 7 (c) 9 (d) 1
3. 3.Which of these numbers is divisible by 9? (a) 6828 (b) 6837 (c) 6842 (d) 6831
4. 4.What is the smallest number you must add to 354 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
5. 5.What is the smallest number you must add to 233 to make it divisible by 6? (a) 5 (b) 1 (c) 3 (d) 2
6. 6.What is the remainder when 88 is divided by 5? (a) 1 (b) 3 (c) 4 (d) 2
7. 7.Which of these numbers is divisible by 4? (a) 650 (b) 652 (c) 659 (d) 655
8. 8.Which of these numbers is divisible by 3? (a) 1765 (b) 1762 (c) 1759 (d) 1761
9. 9.Which of these numbers is divisible by 9? (a) 2818 (b) 2808 (c) 2811 (d) 2800
10. 10.What is the units digit of 4^30? (a) 4 (b) 7 (c) 6 (d) 5
11. 11.What is the units digit of 4^12? (a) 7 (b) 6 (c) 4 (d) 5
12. 12.What is the remainder when 219 is divided by 6? (a) 5 (b) 2 (c) 3 (d) 4
13. 13.What is the remainder when 347 is divided by 8? (a) 3 (b) 5 (c) 2 (d) 4
14. 14.Which of these numbers is divisible by 4? (a) 3458 (b) 3454 (c) 3452 (d) 3450
15. 15.Which of these numbers is divisible by 11? (a) 9763 (b) 9768 (c) 9787 (d) 9769
16. 16.Which of these numbers is divisible by 8? (a) 902 (b) 904 (c) 916 (d) 908
17. 17.What is the smallest number you must add to 110 to make it divisible by 7? (a) 3 (b) 1 (c) 5 (d) 2
18. 18.What is the smallest number you must add to 731 to make it divisible by 10? (a) 8 (b) 7 (c) 1 (d) 9
19. 19.What is the remainder when 392 is divided by 11? (a) 8 (b) 6 (c) 4 (d) 7
20. 20.What is the remainder when 203 is divided by 3? (a) 0 (b) 2 (c) 1
21. 21.What is the units digit of 2^13? (a) 6 (b) 2 (c) 8 (d) 4
22. 22.What is the remainder when 237 is divided by 4? (a) 0 (b) 2 (c) 1 (d) 3
23. 23.Which of these numbers is divisible by 11? (a) 4204 (b) 4191 (c) 4192 (d) 4183
24. 24.Which of these numbers is divisible by 6? (a) 1468 (b) 1463 (c) 1472 (d) 1464
25. 25.What is the smallest number you must add to 370 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 2
26. 26.What is the smallest number you must add to 228 to make it divisible by 5? (a) 3 (b) 2 (c) 1 (d) 4
27. 27.Which of these numbers is divisible by 3? (a) 688 (b) 686 (c) 683 (d) 684
28. 28.What is the remainder when 293 is divided by 8? (a) 3 (b) 4 (c) 6 (d) 5
29. 29.Which of these numbers is divisible by 3? (a) 2038 (b) 2041 (c) 2036 (d) 2037
30. 30.What is the remainder when 112 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
31. 31.Which of these numbers is divisible by 11? (a) 4571 (b) 4562 (c) 4565 (d) 4577
32. 32.What is the smallest number you must add to 77 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
33. 33.What is the remainder when 79 is divided by 3? (a) 2 (b) 0 (c) 1
34. 34.Which of these numbers is divisible by 8? (a) 4300 (b) 4296 (c) 4305 (d) 4291
35. 35.Which of these numbers is divisible by 6? (a) 3012 (b) 3022 (c) 3015 (d) 3007
36. 36.What is the smallest number you must add to 285 to make it divisible by 4? (a) 2 (b) 3 (c) 1
37. 37.What is the remainder when 236 is divided by 5? (a) 4 (b) 0 (c) 1 (d) 2
38. 38.What is the remainder when 181 is divided by 3? (a) 2 (b) 1 (c) 0
39. 39.What is the units digit of 8^56? (a) 8 (b) 2 (c) 6 (d) 4
40. 40.What is the remainder when 111 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2

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