Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 966 · 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 smallest number you must add to 212 to make it divisible by 10? (a) 8 (b) 2 (c) 7 (d) 9
2. 2.Which of these numbers is divisible by 4? (a) 3342 (b) 3349 (c) 3345 (d) 3344
3. 3.What is the units digit of 3^31? (a) 9 (b) 3 (c) 7 (d) 1
4. 4.Which of these numbers is divisible by 8? (a) 6187 (b) 6179 (c) 6196 (d) 6184
5. 5.Which of these numbers is divisible by 11? (a) 9854 (b) 9859 (c) 9835 (d) 9845
6. 6.What is the smallest number you must add to 200 to make it divisible by 3? (a) 1 (b) 2
7. 7.Which of these numbers is divisible by 4? (a) 2459 (b) 2449 (c) 2455 (d) 2452
8. 8.Which of these numbers is divisible by 8? (a) 6886 (b) 6893 (c) 6901 (d) 6888
9. 9.What is the smallest number you must add to 143 to make it divisible by 6? (a) 3 (b) 5 (c) 2 (d) 1
10. 10.Which of these numbers is divisible by 3? (a) 1818 (b) 1820 (c) 1816 (d) 1822
11. 11.Which of these numbers is divisible by 8? (a) 2183 (b) 2190 (c) 2176 (d) 2173
12. 12.What is the smallest number you must add to 282 to make it divisible by 7? (a) 4 (b) 5 (c) 6 (d) 2
13. 13.Which of these numbers is divisible by 6? (a) 3062 (b) 3070 (c) 3060 (d) 3056
14. 14.What is the remainder when 249 is divided by 7? (a) 4 (b) 6 (c) 3 (d) 5
15. 15.What is the units digit of 9^58? (a) 2 (b) 9 (c) 1 (d) 0
16. 16.What is the smallest number you must add to 140 to make it divisible by 12? (a) 8 (b) 4 (c) 3 (d) 5
17. 17.What is the remainder when 176 is divided by 3? (a) 1 (b) 0 (c) 2
18. 18.What is the smallest number you must add to 229 to make it divisible by 3? (a) 2 (b) 1
19. 19.What is the remainder when 542 is divided by 8? (a) 2 (b) 7 (c) 6 (d) 5
20. 20.What is the smallest number you must add to 374 to make it divisible by 6? (a) 3 (b) 4 (c) 2 (d) 5
21. 21.What is the remainder when 55 is divided by 3? (a) 1 (b) 2 (c) 0
22. 22.What is the smallest number you must add to 820 to make it divisible by 12? (a) 8 (b) 4 (c) 9 (d) 7
23. 23.What is the remainder when 162 is divided by 12? (a) 5 (b) 8 (c) 7 (d) 6
24. 24.What is the remainder when 391 is divided by 8? (a) 1 (b) 5 (c) 6 (d) 7
25. 25.Which of these numbers is divisible by 9? (a) 1138 (b) 1148 (c) 1156 (d) 1143
26. 26.What is the smallest number you must add to 166 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
27. 27.What is the units digit of 8^18? (a) 2 (b) 8 (c) 6 (d) 4
28. 28.What is the units digit of 2^59? (a) 6 (b) 8 (c) 4 (d) 2
29. 29.What is the units digit of 4^50? (a) 7 (b) 4 (c) 5 (d) 6
30. 30.What is the units digit of 7^51? (a) 1 (b) 7 (c) 9 (d) 3
31. 31.What is the smallest number you must add to 101 to make it divisible by 6? (a) 1 (b) 5 (c) 2 (d) 3
32. 32.What is the remainder when 715 is divided by 9? (a) 4 (b) 5 (c) 6 (d) 3
33. 33.What is the units digit of 9^16? (a) 9 (b) 2 (c) 0 (d) 1
34. 34.What is the smallest number you must add to 109 to make it divisible by 8? (a) 4 (b) 2 (c) 3 (d) 5
35. 35.Which of these numbers is divisible by 3? (a) 2540 (b) 2535 (c) 2537 (d) 2534
36. 36.What is the remainder when 89 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
37. 37.Which of these numbers is divisible by 8? (a) 1729 (b) 1728 (c) 1739 (d) 1723
38. 38.What is the remainder when 182 is divided by 3? (a) 2 (b) 1 (c) 0
39. 39.What is the units digit of 2^43? (a) 4 (b) 6 (c) 8 (d) 2
40. 40.What is the remainder when 86 is divided by 3? (a) 2 (b) 1 (c) 0

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