Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 411 · 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.Which of these numbers is divisible by 3? (a) 734 (b) 728 (c) 730 (d) 729
2. 2.What is the units digit of 9^13? (a) 8 (b) 9 (c) 7 (d) 1
3. 3.What is the units digit of 3^15? (a) 7 (b) 9 (c) 3 (d) 1
4. 4.Which of these numbers is divisible by 4? (a) 2469 (b) 2468 (c) 2467 (d) 2474
5. 5.What is the remainder when 127 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
6. 6.What is the remainder when 78 is divided by 4? (a) 0 (b) 3 (c) 2 (d) 1
7. 7.What is the remainder when 509 is divided by 12? (a) 6 (b) 4 (c) 5 (d) 7
8. 8.Which of these numbers is divisible by 8? (a) 2660 (b) 2656 (c) 2668 (d) 2652
9. 9.What is the smallest number you must add to 426 to make it divisible by 7? (a) 6 (b) 3 (c) 2 (d) 1
10. 10.What is the remainder when 134 is divided by 3? (a) 1 (b) 0 (c) 2
11. 11.What is the remainder when 88 is divided by 3? (a) 0 (b) 2 (c) 1
12. 12.What is the smallest number you must add to 261 to make it divisible by 7? (a) 6 (b) 2 (c) 4 (d) 5
13. 13.Which of these numbers is divisible by 6? (a) 2991 (b) 3001 (c) 2994 (d) 2995
14. 14.Which of these numbers is divisible by 11? (a) 2999 (b) 2975 (c) 2987 (d) 2981
15. 15.What is the smallest number you must add to 795 to make it divisible by 12? (a) 8 (b) 10 (c) 3 (d) 9
16. 16.Which of these numbers is divisible by 4? (a) 625 (b) 617 (c) 623 (d) 620
17. 17.What is the remainder when 137 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
18. 18.What is the remainder when 742 is divided by 12? (a) 11 (b) 9 (c) 2 (d) 10
19. 19.What is the units digit of 3^35? (a) 3 (b) 9 (c) 1 (d) 7
20. 20.Which of these numbers is divisible by 9? (a) 5321 (b) 5319 (c) 5336 (d) 5317
21. 21.Which of these numbers is divisible by 11? (a) 1043 (b) 1029 (c) 1023 (d) 1015
22. 22.What is the smallest number you must add to 167 to make it divisible by 3? (a) 2 (b) 1
23. 23.What is the remainder when 175 is divided by 6? (a) 0 (b) 2 (c) 1 (d) 5
24. 24.What is the smallest number you must add to 515 to make it divisible by 10? (a) 7 (b) 4 (c) 6 (d) 5
25. 25.What is the smallest number you must add to 329 to make it divisible by 6? (a) 5 (b) 3 (c) 2 (d) 1
26. 26.What is the smallest number you must add to 124 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
27. 27.What is the smallest number you must add to 123 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
28. 28.What is the remainder when 530 is divided by 7? (a) 6 (b) 5 (c) 4 (d) 2
29. 29.What is the units digit of 9^23? (a) 7 (b) 9 (c) 1 (d) 8
30. 30.What is the remainder when 562 is divided by 7? (a) 5 (b) 2 (c) 1 (d) 3
31. 31.What is the smallest number you must add to 438 to make it divisible by 11? (a) 9 (b) 2 (c) 3 (d) 1
32. 32.What is the smallest number you must add to 228 to make it divisible by 9? (a) 7 (b) 5 (c) 3 (d) 6
33. 33.What is the units digit of 4^23? (a) 3 (b) 5 (c) 4 (d) 6
34. 34.What is the remainder when 73 is divided by 3? (a) 0 (b) 2 (c) 1
35. 35.What is the smallest number you must add to 442 to make it divisible by 6? (a) 4 (b) 3 (c) 2 (d) 1
36. 36.What is the remainder when 124 is divided by 6? (a) 5 (b) 4 (c) 3 (d) 2
37. 37.Which of these numbers is divisible by 8? (a) 2260 (b) 2251 (c) 2248 (d) 2247
38. 38.What is the units digit of 8^42? (a) 2 (b) 4 (c) 6 (d) 8
39. 39.What is the remainder when 446 is divided by 7? (a) 6 (b) 5 (c) 2 (d) 4
40. 40.What is the smallest number you must add to 247 to make it divisible by 8? (a) 2 (b) 7 (c) 1 (d) 3

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