Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 460 · 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 remainder when 120 is divided by 7? (a) 0 (b) 1 (c) 6 (d) 2
2. 2.What is the remainder when 191 is divided by 6? (a) 3 (b) 1 (c) 5 (d) 4
3. 3.What is the units digit of 2^59? (a) 6 (b) 2 (c) 8 (d) 4
4. 4.Which of these numbers is divisible by 3? (a) 866 (b) 861 (c) 860 (d) 862
5. 5.What is the smallest number you must add to 334 to make it divisible by 8? (a) 2 (b) 6 (c) 3 (d) 1
6. 6.What is the remainder when 124 is divided by 3? (a) 1 (b) 0 (c) 2
7. 7.What is the smallest number you must add to 314 to make it divisible by 12? (a) 2 (b) 9 (c) 11 (d) 10
8. 8.What is the remainder when 718 is divided by 9? (a) 6 (b) 2 (c) 8 (d) 7
9. 9.Which of these numbers is divisible by 8? (a) 3391 (b) 3376 (c) 3378 (d) 3373
10. 10.What is the units digit of 3^46? (a) 9 (b) 1 (c) 7 (d) 3
11. 11.Which of these numbers is divisible by 8? (a) 4363 (b) 4368 (c) 4372 (d) 4379
12. 12.What is the remainder when 346 is divided by 5? (a) 0 (b) 2 (c) 1 (d) 4
13. 13.What is the units digit of 9^51? (a) 8 (b) 1 (c) 7 (d) 9
14. 14.What is the units digit of 7^18? (a) 1 (b) 3 (c) 9 (d) 7
15. 15.What is the units digit of 7^43? (a) 1 (b) 3 (c) 9 (d) 7
16. 16.What is the remainder when 949 is divided by 12? (a) 1 (b) 0 (c) 2 (d) 11
17. 17.What is the units digit of 8^53? (a) 2 (b) 8 (c) 6 (d) 4
18. 18.What is the smallest number you must add to 511 to make it divisible by 9? (a) 1 (b) 3 (c) 7 (d) 2
19. 19.What is the remainder when 175 is divided by 8? (a) 6 (b) 5 (c) 1 (d) 7
20. 20.What is the remainder when 97 is divided by 7? (a) 5 (b) 1 (c) 4 (d) 6
21. 21.What is the units digit of 2^12? (a) 8 (b) 4 (c) 6 (d) 2
22. 22.What is the remainder when 377 is divided by 5? (a) 4 (b) 1 (c) 2 (d) 3
23. 23.What is the smallest number you must add to 91 to make it divisible by 6? (a) 1 (b) 4 (c) 5 (d) 3
24. 24.What is the remainder when 406 is divided by 8? (a) 2 (b) 5 (c) 6 (d) 7
25. 25.What is the remainder when 121 is divided by 5? (a) 1 (b) 4 (c) 2 (d) 0
26. 26.What is the remainder when 206 is divided by 5? (a) 0 (b) 1 (c) 4 (d) 2
27. 27.What is the smallest number you must add to 608 to make it divisible by 10? (a) 1 (b) 2 (c) 8 (d) 3
28. 28.What is the units digit of 4^13? (a) 3 (b) 5 (c) 4 (d) 6
29. 29.Which of these numbers is divisible by 3? (a) 2186 (b) 2183 (c) 2180 (d) 2181
30. 30.What is the units digit of 8^28? (a) 4 (b) 8 (c) 6 (d) 2
31. 31.What is the remainder when 545 is divided by 9? (a) 5 (b) 7 (c) 6 (d) 4
32. 32.What is the remainder when 317 is divided by 7? (a) 1 (b) 3 (c) 2 (d) 5
33. 33.What is the smallest number you must add to 221 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
34. 34.What is the units digit of 9^25? (a) 9 (b) 8 (c) 1 (d) 7
35. 35.What is the smallest number you must add to 275 to make it divisible by 7? (a) 2 (b) 4 (c) 6 (d) 5
36. 36.What is the remainder when 93 is divided by 4? (a) 0 (b) 3 (c) 1 (d) 2
37. 37.What is the units digit of 8^58? (a) 8 (b) 2 (c) 4 (d) 6
38. 38.What is the smallest number you must add to 128 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
39. 39.What is the smallest number you must add to 307 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
40. 40.What is the units digit of 9^43? (a) 7 (b) 1 (c) 8 (d) 9

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