Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 875 · 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 475 is divided by 11? (a) 3 (b) 9 (c) 1 (d) 2
2. 2.What is the smallest number you must add to 585 to make it divisible by 12? (a) 3 (b) 9 (c) 2 (d) 4
3. 3.What is the units digit of 3^16? (a) 3 (b) 7 (c) 1 (d) 9
4. 4.What is the smallest number you must add to 383 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
5. 5.What is the units digit of 7^26? (a) 1 (b) 7 (c) 9 (d) 3
6. 6.What is the remainder when 91 is divided by 5? (a) 4 (b) 1 (c) 0 (d) 2
7. 7.What is the smallest number you must add to 701 to make it divisible by 12? (a) 8 (b) 5 (c) 7 (d) 6
8. 8.What is the remainder when 78 is divided by 4? (a) 3 (b) 1 (c) 0 (d) 2
9. 9.What is the smallest number you must add to 774 to make it divisible by 11? (a) 7 (b) 8 (c) 6 (d) 4
10. 10.What is the units digit of 9^51? (a) 1 (b) 7 (c) 9 (d) 8
11. 11.What is the remainder when 450 is divided by 12? (a) 8 (b) 5 (c) 6 (d) 7
12. 12.What is the smallest number you must add to 785 to make it divisible by 11? (a) 8 (b) 6 (c) 4 (d) 7
13. 13.What is the units digit of 8^44? (a) 8 (b) 2 (c) 6 (d) 4
14. 14.Which of these numbers is divisible by 4? (a) 2726 (b) 2735 (c) 2731 (d) 2728
15. 15.What is the smallest number you must add to 377 to make it divisible by 8? (a) 1 (b) 6 (c) 5 (d) 7
16. 16.What is the remainder when 312 is divided by 9? (a) 3 (b) 6 (c) 5 (d) 7
17. 17.What is the remainder when 301 is divided by 11? (a) 3 (b) 5 (c) 4 (d) 7
18. 18.What is the remainder when 121 is divided by 9? (a) 6 (b) 5 (c) 3 (d) 4
19. 19.Which of these numbers is divisible by 9? (a) 991 (b) 990 (c) 1000 (d) 986
20. 20.What is the units digit of 9^11? (a) 7 (b) 9 (c) 1 (d) 8
21. 21.Which of these numbers is divisible by 6? (a) 316 (b) 323 (c) 329 (d) 318
22. 22.Which of these numbers is divisible by 9? (a) 4506 (b) 4516 (c) 4497 (d) 4500
23. 23.What is the remainder when 261 is divided by 4? (a) 3 (b) 0 (c) 2 (d) 1
24. 24.What is the smallest number you must add to 135 to make it divisible by 6? (a) 4 (b) 5 (c) 2 (d) 3
25. 25.What is the remainder when 163 is divided by 4? (a) 0 (b) 1 (c) 2 (d) 3
26. 26.Which of these numbers is divisible by 9? (a) 3884 (b) 3893 (c) 3874 (d) 3879
27. 27.What is the remainder when 282 is divided by 4? (a) 3 (b) 0 (c) 1 (d) 2
28. 28.What is the remainder when 80 is divided by 3? (a) 0 (b) 1 (c) 2
29. 29.Which of these numbers is divisible by 9? (a) 1981 (b) 1996 (c) 1977 (d) 1980
30. 30.Which of these numbers is divisible by 9? (a) 3307 (b) 3325 (c) 3314 (d) 3312
31. 31.Which of these numbers is divisible by 8? (a) 3540 (b) 3554 (c) 3544 (d) 3551
32. 32.Which of these numbers is divisible by 3? (a) 1664 (b) 1670 (c) 1667 (d) 1665
33. 33.What is the units digit of 2^55? (a) 2 (b) 4 (c) 6 (d) 8
34. 34.What is the remainder when 79 is divided by 3? (a) 0 (b) 2 (c) 1
35. 35.What is the smallest number you must add to 471 to make it divisible by 7? (a) 2 (b) 5 (c) 6 (d) 4
36. 36.What is the remainder when 283 is divided by 12? (a) 5 (b) 6 (c) 7 (d) 8
37. 37.What is the units digit of 7^20? (a) 3 (b) 9 (c) 7 (d) 1
38. 38.What is the units digit of 3^53? (a) 1 (b) 7 (c) 3 (d) 9
39. 39.What is the remainder when 220 is divided by 12? (a) 3 (b) 4 (c) 5 (d) 8
40. 40.What is the smallest number you must add to 782 to make it divisible by 10? (a) 7 (b) 8 (c) 9 (d) 2

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