Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 837 · 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) 1522 (b) 1520 (c) 1526 (d) 1521
2. 2.What is the units digit of 8^45? (a) 2 (b) 6 (c) 4 (d) 8
3. 3.What is the remainder when 205 is divided by 7? (a) 3 (b) 5 (c) 1 (d) 2
4. 4.What is the smallest number you must add to 367 to make it divisible by 11? (a) 7 (b) 6 (c) 4 (d) 8
5. 5.What is the units digit of 3^51? (a) 7 (b) 3 (c) 1 (d) 9
6. 6.What is the smallest number you must add to 615 to make it divisible by 11? (a) 1 (b) 3 (c) 10 (d) 2
7. 7.What is the units digit of 3^39? (a) 1 (b) 9 (c) 3 (d) 7
8. 8.What is the remainder when 74 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
9. 9.What is the units digit of 2^14? (a) 8 (b) 2 (c) 4 (d) 6
10. 10.What is the remainder when 383 is divided by 11? (a) 2 (b) 10 (c) 9 (d) 8
11. 11.What is the units digit of 9^41? (a) 9 (b) 7 (c) 8 (d) 1
12. 12.What is the units digit of 4^44? (a) 7 (b) 5 (c) 6 (d) 4
13. 13.What is the remainder when 334 is divided by 10? (a) 6 (b) 5 (c) 3 (d) 4
14. 14.What is the units digit of 3^38? (a) 1 (b) 3 (c) 7 (d) 9
15. 15.What is the units digit of 7^20? (a) 3 (b) 1 (c) 7 (d) 9
16. 16.What is the units digit of 7^39? (a) 3 (b) 1 (c) 9 (d) 7
17. 17.Which of these numbers is divisible by 4? (a) 696 (b) 702 (c) 695 (d) 697
18. 18.What is the units digit of 7^37? (a) 1 (b) 9 (c) 7 (d) 3
19. 19.What is the smallest number you must add to 250 to make it divisible by 6? (a) 3 (b) 1 (c) 2 (d) 4
20. 20.What is the units digit of 3^12? (a) 3 (b) 7 (c) 1 (d) 9
21. 21.What is the smallest number you must add to 647 to make it divisible by 10? (a) 4 (b) 2 (c) 3 (d) 7
22. 22.What is the remainder when 583 is divided by 8? (a) 1 (b) 6 (c) 7 (d) 5
23. 23.What is the remainder when 627 is divided by 12? (a) 2 (b) 9 (c) 3 (d) 4
24. 24.What is the remainder when 202 is divided by 8? (a) 2 (b) 6 (c) 1 (d) 3
25. 25.What is the smallest number you must add to 455 to make it divisible by 6? (a) 2 (b) 1 (c) 3 (d) 5
26. 26.What is the smallest number you must add to 390 to make it divisible by 12? (a) 5 (b) 7 (c) 8 (d) 6
27. 27.What is the smallest number you must add to 635 to make it divisible by 8? (a) 6 (b) 4 (c) 5 (d) 3
28. 28.What is the remainder when 465 is divided by 6? (a) 3 (b) 4 (c) 5 (d) 2
29. 29.Which of these numbers is divisible by 3? (a) 604 (b) 610 (c) 606 (d) 607
30. 30.What is the smallest number you must add to 463 to make it divisible by 10? (a) 6 (b) 8 (c) 7 (d) 3
31. 31.What is the smallest number you must add to 746 to make it divisible by 12? (a) 11 (b) 2 (c) 9 (d) 10
32. 32.Which of these numbers is divisible by 9? (a) 7480 (b) 7470 (c) 7468 (d) 7477
33. 33.Which of these numbers is divisible by 4? (a) 250 (b) 244 (c) 247 (d) 243
34. 34.What is the remainder when 598 is divided by 10? (a) 7 (b) 9 (c) 8 (d) 2
35. 35.What is the remainder when 197 is divided by 9? (a) 8 (b) 7 (c) 1 (d) 6
36. 36.What is the remainder when 114 is divided by 9? (a) 5 (b) 3 (c) 7 (d) 6
37. 37.Which of these numbers is divisible by 6? (a) 2089 (b) 2095 (c) 2101 (d) 2094
38. 38.What is the remainder when 263 is divided by 4? (a) 1 (b) 2 (c) 0 (d) 3
39. 39.What is the remainder when 524 is divided by 7? (a) 5 (b) 1 (c) 4 (d) 6
40. 40.What is the smallest number you must add to 248 to make it divisible by 7? (a) 3 (b) 4 (c) 5 (d) 6

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