Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 880 · 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.Which of these numbers is divisible by 8? (a) 4859 (b) 4850 (c) 4846 (d) 4848
2. 2.What is the remainder when 249 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
3. 3.What is the smallest number you must add to 176 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
4. 4.What is the units digit of 4^58? (a) 5 (b) 6 (c) 4 (d) 7
5. 5.Which of these numbers is divisible by 4? (a) 558 (b) 549 (c) 552 (d) 554
6. 6.What is the smallest number you must add to 142 to make it divisible by 8? (a) 2 (b) 3 (c) 6 (d) 1
7. 7.What is the remainder when 392 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
8. 8.What is the units digit of 7^50? (a) 7 (b) 3 (c) 1 (d) 9
9. 9.Which of these numbers is divisible by 3? (a) 1801 (b) 1798 (c) 1795 (d) 1797
10. 10.What is the remainder when 578 is divided by 9? (a) 1 (b) 7 (c) 3 (d) 2
11. 11.What is the units digit of 4^12? (a) 4 (b) 6 (c) 5 (d) 7
12. 12.What is the remainder when 585 is divided by 8? (a) 7 (b) 0 (c) 1 (d) 2
13. 13.What is the remainder when 225 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
14. 14.Which of these numbers is divisible by 3? (a) 919 (b) 917 (c) 915 (d) 913
15. 15.What is the smallest number you must add to 408 to make it divisible by 11? (a) 8 (b) 9 (c) 10 (d) 1
16. 16.Which of these numbers is divisible by 4? (a) 1782 (b) 1779 (c) 1776 (d) 1774
17. 17.Which of these numbers is divisible by 11? (a) 5555 (b) 5568 (c) 5547 (d) 5558
18. 18.What is the smallest number you must add to 203 to make it divisible by 11? (a) 6 (b) 5 (c) 7 (d) 8
19. 19.What is the units digit of 7^27? (a) 1 (b) 3 (c) 7 (d) 9
20. 20.Which of these numbers is divisible by 3? (a) 949 (b) 955 (c) 951 (d) 952
21. 21.Which of these numbers is divisible by 8? (a) 3216 (b) 3212 (c) 3229 (d) 3222
22. 22.What is the remainder when 387 is divided by 7? (a) 1 (b) 3 (c) 2 (d) 5
23. 23.What is the remainder when 447 is divided by 6? (a) 5 (b) 2 (c) 4 (d) 3
24. 24.What is the smallest number you must add to 138 to make it divisible by 7? (a) 5 (b) 1 (c) 3 (d) 2
25. 25.What is the smallest number you must add to 735 to make it divisible by 10? (a) 5 (b) 6 (c) 4 (d) 7
26. 26.What is the smallest number you must add to 74 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
27. 27.What is the units digit of 9^20? (a) 2 (b) 0 (c) 9 (d) 1
28. 28.What is the remainder when 310 is divided by 7? (a) 5 (b) 1 (c) 2 (d) 3
29. 29.What is the remainder when 149 is divided by 8? (a) 6 (b) 3 (c) 5 (d) 4
30. 30.Which of these numbers is divisible by 3? (a) 1102 (b) 1104 (c) 1109 (d) 1106
31. 31.What is the units digit of 7^40? (a) 7 (b) 1 (c) 3 (d) 9
32. 32.Which of these numbers is divisible by 11? (a) 5045 (b) 5050 (c) 5038 (d) 5032
33. 33.What is the smallest number you must add to 637 to make it divisible by 9? (a) 2 (b) 3 (c) 7 (d) 1
34. 34.What is the smallest number you must add to 872 to make it divisible by 11? (a) 3 (b) 7 (c) 8 (d) 9
35. 35.What is the units digit of 3^33? (a) 7 (b) 1 (c) 9 (d) 3
36. 36.What is the remainder when 565 is divided by 11? (a) 7 (b) 4 (c) 3 (d) 5
37. 37.Which of these numbers is divisible by 9? (a) 5415 (b) 5400 (c) 5401 (d) 5395
38. 38.Which of these numbers is divisible by 6? (a) 1832 (b) 1847 (c) 1840 (d) 1836
39. 39.What is the units digit of 2^12? (a) 6 (b) 4 (c) 2 (d) 8
40. 40.What is the units digit of 3^25? (a) 1 (b) 7 (c) 9 (d) 3

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