Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 105 · 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 199 to make it divisible by 8? (a) 1 (b) 3 (c) 7 (d) 2
2. 2.Which of these numbers is divisible by 9? (a) 2568 (b) 2560 (c) 2565 (d) 2577
3. 3.Which of these numbers is divisible by 9? (a) 2042 (b) 2051 (c) 2034 (d) 2032
4. 4.What is the smallest number you must add to 670 to make it divisible by 11? (a) 3 (b) 2 (c) 10 (d) 1
5. 5.What is the units digit of 8^16? (a) 2 (b) 4 (c) 8 (d) 6
6. 6.What is the smallest number you must add to 165 to make it divisible by 10? (a) 7 (b) 6 (c) 5 (d) 4
7. 7.What is the units digit of 3^15? (a) 7 (b) 9 (c) 3 (d) 1
8. 8.Which of these numbers is divisible by 8? (a) 2235 (b) 2221 (c) 2226 (d) 2224
9. 9.Which of these numbers is divisible by 8? (a) 4045 (b) 4049 (c) 4058 (d) 4048
10. 10.What is the remainder when 73 is divided by 3? (a) 2 (b) 0 (c) 1
11. 11.What is the remainder when 508 is divided by 8? (a) 5 (b) 4 (c) 6 (d) 3
12. 12.What is the units digit of 3^25? (a) 9 (b) 3 (c) 1 (d) 7
13. 13.Which of these numbers is divisible by 4? (a) 1828 (b) 1827 (c) 1834 (d) 1829
14. 14.What is the remainder when 138 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
15. 15.What is the smallest number you must add to 89 to make it divisible by 3? (a) 2 (b) 1
16. 16.What is the units digit of 4^27? (a) 3 (b) 6 (c) 5 (d) 4
17. 17.What is the remainder when 639 is divided by 12? (a) 9 (b) 2 (c) 3 (d) 4
18. 18.What is the remainder when 323 is divided by 7? (a) 2 (b) 0 (c) 6 (d) 1
19. 19.What is the remainder when 548 is divided by 10? (a) 7 (b) 8 (c) 9 (d) 2
20. 20.Which of these numbers is divisible by 3? (a) 1994 (b) 1990 (c) 1992 (d) 1997
21. 21.What is the remainder when 196 is divided by 6? (a) 5 (b) 4 (c) 2 (d) 3
22. 22.What is the remainder when 115 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
23. 23.Which of these numbers is divisible by 9? (a) 7192 (b) 7189 (c) 7191 (d) 7205
24. 24.What is the smallest number you must add to 290 to make it divisible by 8? (a) 6 (b) 7 (c) 5 (d) 2
25. 25.What is the remainder when 303 is divided by 10? (a) 7 (b) 2 (c) 3 (d) 4
26. 26.What is the units digit of 4^19? (a) 3 (b) 6 (c) 4 (d) 5
27. 27.What is the remainder when 97 is divided by 6? (a) 2 (b) 1 (c) 0 (d) 5
28. 28.Which of these numbers is divisible by 11? (a) 1311 (b) 1333 (c) 1326 (d) 1320
29. 29.What is the remainder when 721 is divided by 11? (a) 8 (b) 5 (c) 6 (d) 7
30. 30.What is the smallest number you must add to 247 to make it divisible by 4? (a) 1 (b) 2 (c) 3
31. 31.What is the smallest number you must add to 607 to make it divisible by 8? (a) 3 (b) 2 (c) 7 (d) 1
32. 32.What is the remainder when 56 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 0
33. 33.What is the remainder when 417 is divided by 10? (a) 3 (b) 8 (c) 6 (d) 7
34. 34.What is the smallest number you must add to 151 to make it divisible by 3? (a) 2 (b) 1
35. 35.Which of these numbers is divisible by 9? (a) 5398 (b) 5404 (c) 5389 (d) 5391
36. 36.Which of these numbers is divisible by 8? (a) 1721 (b) 1712 (c) 1717 (d) 1711
37. 37.What is the units digit of 8^47? (a) 4 (b) 2 (c) 8 (d) 6
38. 38.What is the remainder when 365 is divided by 7? (a) 6 (b) 2 (c) 0 (d) 1
39. 39.What is the smallest number you must add to 456 to make it divisible by 10? (a) 6 (b) 4 (c) 3 (d) 5
40. 40.Which of these numbers is divisible by 11? (a) 3051 (b) 3058 (c) 3066 (d) 3079

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