Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 395 · 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 6? (a) 746 (b) 744 (c) 739 (d) 754
2. 2.What is the units digit of 9^10? (a) 9 (b) 0 (c) 2 (d) 1
3. 3.What is the remainder when 262 is divided by 7? (a) 5 (b) 3 (c) 2 (d) 4
4. 4.What is the remainder when 208 is divided by 9? (a) 0 (b) 1 (c) 2 (d) 8
5. 5.What is the remainder when 115 is divided by 9? (a) 7 (b) 8 (c) 2 (d) 6
6. 6.What is the units digit of 8^11? (a) 2 (b) 4 (c) 8 (d) 6
7. 7.What is the smallest number you must add to 49 to make it divisible by 3? (a) 2 (b) 1
8. 8.Which of these numbers is divisible by 9? (a) 1801 (b) 1812 (c) 1809 (d) 1823
9. 9.What is the smallest number you must add to 453 to make it divisible by 9? (a) 6 (b) 7 (c) 5 (d) 3
10. 10.What is the units digit of 3^56? (a) 1 (b) 3 (c) 9 (d) 7
11. 11.Which of these numbers is divisible by 3? (a) 2657 (b) 2653 (c) 2655 (d) 2659
12. 12.Which of these numbers is divisible by 3? (a) 895 (b) 890 (c) 893 (d) 891
13. 13.Which of these numbers is divisible by 3? (a) 1990 (b) 1994 (c) 1992 (d) 1996
14. 14.What is the smallest number you must add to 383 to make it divisible by 12? (a) 11 (b) 1 (c) 2 (d) 3
15. 15.What is the remainder when 932 is divided by 12? (a) 9 (b) 4 (c) 8 (d) 7
16. 16.What is the units digit of 3^31? (a) 1 (b) 7 (c) 9 (d) 3
17. 17.What is the remainder when 182 is divided by 9? (a) 7 (b) 2 (c) 3 (d) 1
18. 18.What is the smallest number you must add to 545 to make it divisible by 12? (a) 8 (b) 7 (c) 6 (d) 5
19. 19.Which of these numbers is divisible by 4? (a) 1395 (b) 1391 (c) 1387 (d) 1388
20. 20.What is the smallest number you must add to 596 to make it divisible by 12? (a) 4 (b) 3 (c) 5 (d) 8
21. 21.What is the remainder when 73 is divided by 4? (a) 3 (b) 2 (c) 0 (d) 1
22. 22.What is the units digit of 3^44? (a) 3 (b) 1 (c) 7 (d) 9
23. 23.What is the units digit of 4^23? (a) 5 (b) 3 (c) 4 (d) 6
24. 24.What is the units digit of 8^49? (a) 4 (b) 8 (c) 6 (d) 2
25. 25.What is the units digit of 8^40? (a) 4 (b) 2 (c) 6 (d) 8
26. 26.What is the units digit of 9^39? (a) 1 (b) 8 (c) 9 (d) 7
27. 27.What is the smallest number you must add to 621 to make it divisible by 10? (a) 9 (b) 7 (c) 1 (d) 8
28. 28.What is the remainder when 300 is divided by 8? (a) 6 (b) 5 (c) 3 (d) 4
29. 29.What is the remainder when 627 is divided by 10? (a) 7 (b) 8 (c) 3 (d) 6
30. 30.What is the units digit of 3^12? (a) 7 (b) 1 (c) 3 (d) 9
31. 31.What is the remainder when 202 is divided by 3? (a) 0 (b) 1 (c) 2
32. 32.What is the units digit of 3^39? (a) 3 (b) 7 (c) 1 (d) 9
33. 33.What is the smallest number you must add to 107 to make it divisible by 7? (a) 4 (b) 6 (c) 5 (d) 2
34. 34.Which of these numbers is divisible by 6? (a) 714 (b) 723 (c) 712 (d) 716
35. 35.What is the smallest number you must add to 54 to make it divisible by 4? (a) 2 (b) 1 (c) 3
36. 36.What is the remainder when 510 is divided by 8? (a) 6 (b) 5 (c) 7 (d) 2
37. 37.What is the smallest number you must add to 925 to make it divisible by 12? (a) 11 (b) 10 (c) 1 (d) 9
38. 38.What is the remainder when 636 is divided by 8? (a) 6 (b) 3 (c) 4 (d) 5
39. 39.Which of these numbers is divisible by 8? (a) 5333 (b) 5339 (c) 5345 (d) 5336
40. 40.What is the smallest number you must add to 263 to make it divisible by 5? (a) 4 (b) 2 (c) 3 (d) 1

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