Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 297 · 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 209 to make it divisible by 7? (a) 6 (b) 2 (c) 3 (d) 1
2. 2.What is the remainder when 258 is divided by 8? (a) 1 (b) 6 (c) 2 (d) 3
3. 3.What is the remainder when 281 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
4. 4.Which of these numbers is divisible by 3? (a) 311 (b) 308 (c) 313 (d) 309
5. 5.What is the remainder when 160 is divided by 3? (a) 1 (b) 0 (c) 2
6. 6.What is the smallest number you must add to 263 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1
7. 7.What is the units digit of 3^27? (a) 3 (b) 1 (c) 9 (d) 7
8. 8.What is the smallest number you must add to 238 to make it divisible by 3? (a) 1 (b) 2
9. 9.What is the units digit of 3^57? (a) 9 (b) 1 (c) 7 (d) 3
10. 10.What is the remainder when 629 is divided by 8? (a) 6 (b) 3 (c) 4 (d) 5
11. 11.What is the smallest number you must add to 53 to make it divisible by 4? (a) 3 (b) 2 (c) 1
12. 12.What is the units digit of 9^29? (a) 9 (b) 8 (c) 1 (d) 7
13. 13.Which of these numbers is divisible by 3? (a) 2179 (b) 2181 (c) 2183 (d) 2186
14. 14.Which of these numbers is divisible by 8? (a) 933 (b) 919 (c) 921 (d) 920
15. 15.What is the remainder when 229 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
16. 16.What is the smallest number you must add to 271 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 5
17. 17.Which of these numbers is divisible by 4? (a) 1982 (b) 1979 (c) 1986 (d) 1980
18. 18.Which of these numbers is divisible by 8? (a) 6477 (b) 6467 (c) 6464 (d) 6463
19. 19.What is the remainder when 247 is divided by 6? (a) 5 (b) 0 (c) 2 (d) 1
20. 20.What is the smallest number you must add to 366 to make it divisible by 8? (a) 3 (b) 1 (c) 2 (d) 6
21. 21.What is the remainder when 133 is divided by 3? (a) 0 (b) 2 (c) 1
22. 22.What is the smallest number you must add to 459 to make it divisible by 10? (a) 3 (b) 1 (c) 2 (d) 9
23. 23.What is the smallest number you must add to 628 to make it divisible by 9? (a) 1 (b) 3 (c) 7 (d) 2
24. 24.What is the units digit of 7^30? (a) 3 (b) 1 (c) 9 (d) 7
25. 25.What is the smallest number you must add to 405 to make it divisible by 10? (a) 4 (b) 6 (c) 7 (d) 5
26. 26.Which of these numbers is divisible by 8? (a) 6126 (b) 6135 (c) 6120 (d) 6118
27. 27.What is the smallest number you must add to 305 to make it divisible by 4? (a) 3 (b) 1 (c) 2
28. 28.What is the units digit of 4^11? (a) 5 (b) 4 (c) 6 (d) 3
29. 29.What is the units digit of 9^18? (a) 1 (b) 0 (c) 2 (d) 9
30. 30.Which of these numbers is divisible by 8? (a) 913 (b) 927 (c) 905 (d) 912
31. 31.Which of these numbers is divisible by 11? (a) 2860 (b) 2868 (c) 2851 (d) 2875
32. 32.What is the remainder when 323 is divided by 5? (a) 1 (b) 2 (c) 3 (d) 4
33. 33.Which of these numbers is divisible by 11? (a) 5921 (b) 5918 (c) 5933 (d) 5917
34. 34.What is the remainder when 309 is divided by 7? (a) 1 (b) 6 (c) 2 (d) 0
35. 35.What is the smallest number you must add to 346 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
36. 36.What is the units digit of 7^17? (a) 7 (b) 1 (c) 3 (d) 9
37. 37.What is the remainder when 58 is divided by 3? (a) 2 (b) 1 (c) 0
38. 38.Which of these numbers is divisible by 11? (a) 487 (b) 495 (c) 504 (d) 514
39. 39.Which of these numbers is divisible by 6? (a) 3927 (b) 3924 (c) 3920 (d) 3933
40. 40.What is the smallest number you must add to 329 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1

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