Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 619 · 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 9^48? (a) 9 (b) 0 (c) 2 (d) 1
2. 2.What is the units digit of 2^27? (a) 4 (b) 6 (c) 8 (d) 2
3. 3.Which of these numbers is divisible by 4? (a) 514 (b) 516 (c) 518 (d) 523
4. 4.What is the remainder when 242 is divided by 3? (a) 2 (b) 1 (c) 0
5. 5.What is the smallest number you must add to 502 to make it divisible by 9? (a) 7 (b) 1 (c) 2 (d) 3
6. 6.What is the smallest number you must add to 329 to make it divisible by 10? (a) 9 (b) 3 (c) 2 (d) 1
7. 7.What is the remainder when 152 is divided by 3? (a) 2 (b) 0 (c) 1
8. 8.Which of these numbers is divisible by 4? (a) 315 (b) 322 (c) 318 (d) 316
9. 9.What is the smallest number you must add to 321 to make it divisible by 12? (a) 4 (b) 9 (c) 3 (d) 2
10. 10.What is the remainder when 229 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
11. 11.What is the units digit of 9^15? (a) 9 (b) 8 (c) 7 (d) 1
12. 12.Which of these numbers is divisible by 11? (a) 4913 (b) 4898 (c) 4895 (d) 4887
13. 13.What is the smallest number you must add to 189 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
14. 14.What is the remainder when 230 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
15. 15.What is the remainder when 35 is divided by 3? (a) 0 (b) 2 (c) 1
16. 16.What is the units digit of 7^27? (a) 1 (b) 7 (c) 9 (d) 3
17. 17.What is the remainder when 170 is divided by 6? (a) 4 (b) 2 (c) 1 (d) 3
18. 18.What is the remainder when 500 is divided by 7? (a) 2 (b) 4 (c) 5 (d) 3
19. 19.What is the units digit of 3^12? (a) 3 (b) 1 (c) 9 (d) 7
20. 20.What is the remainder when 164 is divided by 6? (a) 3 (b) 4 (c) 2 (d) 1
21. 21.What is the remainder when 171 is divided by 10? (a) 9 (b) 2 (c) 0 (d) 1
22. 22.What is the remainder when 143 is divided by 7? (a) 2 (b) 4 (c) 3 (d) 5
23. 23.What is the smallest number you must add to 335 to make it divisible by 7? (a) 6 (b) 1 (c) 3 (d) 2
24. 24.Which of these numbers is divisible by 6? (a) 4344 (b) 4353 (c) 4340 (d) 4345
25. 25.What is the smallest number you must add to 843 to make it divisible by 12? (a) 8 (b) 10 (c) 3 (d) 9
26. 26.What is the smallest number you must add to 610 to make it divisible by 11? (a) 6 (b) 7 (c) 8 (d) 5
27. 27.What is the smallest number you must add to 126 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
28. 28.Which of these numbers is divisible by 8? (a) 2120 (b) 2135 (c) 2119 (d) 2122
29. 29.What is the remainder when 223 is divided by 3? (a) 2 (b) 1 (c) 0
30. 30.Which of these numbers is divisible by 6? (a) 4015 (b) 4024 (c) 4020 (d) 4029
31. 31.What is the smallest number you must add to 209 to make it divisible by 3? (a) 1 (b) 2
32. 32.What is the units digit of 7^38? (a) 1 (b) 3 (c) 9 (d) 7
33. 33.What is the smallest number you must add to 643 to make it divisible by 11? (a) 5 (b) 7 (c) 6 (d) 8
34. 34.What is the smallest number you must add to 70 to make it divisible by 3? (a) 1 (b) 2
35. 35.Which of these numbers is divisible by 9? (a) 4455 (b) 4466 (c) 4448 (d) 4461
36. 36.What is the units digit of 4^11? (a) 4 (b) 3 (c) 5 (d) 6
37. 37.Which of these numbers is divisible by 3? (a) 2154 (b) 2156 (c) 2158 (d) 2153
38. 38.Which of these numbers is divisible by 3? (a) 172 (b) 174 (c) 179 (d) 175
39. 39.What is the remainder when 436 is divided by 12? (a) 8 (b) 4 (c) 5 (d) 3
40. 40.What is the remainder when 444 is divided by 11? (a) 7 (b) 5 (c) 4 (d) 3

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