Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 37 · 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 226 to make it divisible by 3? (a) 1 (b) 2
2. 2.What is the remainder when 620 is divided by 8? (a) 3 (b) 6 (c) 4 (d) 5
3. 3.Which of these numbers is divisible by 8? (a) 5986 (b) 5976 (c) 5970 (d) 5982
4. 4.What is the remainder when 677 is divided by 11? (a) 7 (b) 8 (c) 6 (d) 5
5. 5.Which of these numbers is divisible by 3? (a) 611 (b) 609 (c) 608 (d) 614
6. 6.What is the smallest number you must add to 369 to make it divisible by 7? (a) 1 (b) 2 (c) 5 (d) 3
7. 7.What is the units digit of 8^55? (a) 8 (b) 4 (c) 2 (d) 6
8. 8.Which of these numbers is divisible by 6? (a) 756 (b) 764 (c) 753 (d) 760
9. 9.What is the units digit of 3^26? (a) 9 (b) 3 (c) 7 (d) 1
10. 10.What is the smallest number you must add to 650 to make it divisible by 11? (a) 1 (b) 8 (c) 9 (d) 10
11. 11.What is the units digit of 8^11? (a) 6 (b) 4 (c) 2 (d) 8
12. 12.What is the smallest number you must add to 51 to make it divisible by 4? (a) 1 (b) 3 (c) 2
13. 13.What is the smallest number you must add to 293 to make it divisible by 4? (a) 1 (b) 3 (c) 2
14. 14.What is the units digit of 2^18? (a) 2 (b) 6 (c) 4 (d) 8
15. 15.Which of these numbers is divisible by 6? (a) 2647 (b) 2635 (c) 2640 (d) 2645
16. 16.What is the remainder when 431 is divided by 7? (a) 4 (b) 6 (c) 3 (d) 5
17. 17.What is the smallest number you must add to 225 to make it divisible by 10? (a) 4 (b) 5 (c) 6 (d) 7
18. 18.What is the units digit of 8^24? (a) 6 (b) 2 (c) 4 (d) 8
19. 19.What is the remainder when 178 is divided by 7? (a) 2 (b) 4 (c) 3 (d) 5
20. 20.Which of these numbers is divisible by 9? (a) 7818 (b) 7826 (c) 7834 (d) 7821
21. 21.What is the units digit of 7^51? (a) 1 (b) 7 (c) 3 (d) 9
22. 22.What is the smallest number you must add to 78 to make it divisible by 7? (a) 1 (b) 6 (c) 5 (d) 4
23. 23.Which of these numbers is divisible by 6? (a) 1551 (b) 1563 (c) 1559 (d) 1554
24. 24.Which of these numbers is divisible by 6? (a) 4455 (b) 4444 (c) 4446 (d) 4449
25. 25.What is the remainder when 49 is divided by 3? (a) 1 (b) 2 (c) 0
26. 26.Which of these numbers is divisible by 8? (a) 4093 (b) 4087 (c) 4080 (d) 4075
27. 27.What is the units digit of 2^36? (a) 2 (b) 8 (c) 6 (d) 4
28. 28.Which of these numbers is divisible by 8? (a) 6090 (b) 6100 (c) 6088 (d) 6082
29. 29.What is the smallest number you must add to 239 to make it divisible by 11? (a) 4 (b) 2 (c) 8 (d) 3
30. 30.What is the smallest number you must add to 171 to make it divisible by 4? (a) 2 (b) 1 (c) 3
31. 31.What is the remainder when 172 is divided by 8? (a) 6 (b) 4 (c) 3 (d) 5
32. 32.What is the smallest number you must add to 241 to make it divisible by 11? (a) 10 (b) 1 (c) 2 (d) 3
33. 33.What is the smallest number you must add to 478 to make it divisible by 8? (a) 3 (b) 6 (c) 1 (d) 2
34. 34.Which of these numbers is divisible by 3? (a) 1170 (b) 1171 (c) 1174 (d) 1168
35. 35.What is the smallest number you must add to 61 to make it divisible by 3? (a) 2 (b) 1
36. 36.What is the remainder when 415 is divided by 6? (a) 1 (b) 5 (c) 2 (d) 0
37. 37.Which of these numbers is divisible by 6? (a) 406 (b) 401 (c) 394 (d) 396
38. 38.What is the remainder when 752 is divided by 11? (a) 5 (b) 7 (c) 3 (d) 4
39. 39.What is the remainder when 392 is divided by 5? (a) 3 (b) 4 (c) 2 (d) 1
40. 40.What is the units digit of 2^15? (a) 4 (b) 2 (c) 6 (d) 8

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