Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 92 · 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 units digit of 7^57? (a) 1 (b) 9 (c) 3 (d) 7
2. 2.What is the units digit of 3^20? (a) 7 (b) 1 (c) 9 (d) 3
3. 3.What is the smallest number you must add to 511 to make it divisible by 10? (a) 9 (b) 7 (c) 8 (d) 1
4. 4.What is the units digit of 4^19? (a) 3 (b) 4 (c) 5 (d) 6
5. 5.What is the smallest number you must add to 277 to make it divisible by 11? (a) 9 (b) 2 (c) 10 (d) 8
6. 6.What is the units digit of 4^12? (a) 7 (b) 6 (c) 4 (d) 5
7. 7.What is the smallest number you must add to 482 to make it divisible by 7? (a) 6 (b) 3 (c) 1 (d) 2
8. 8.What is the remainder when 307 is divided by 8? (a) 5 (b) 2 (c) 3 (d) 4
9. 9.Which of these numbers is divisible by 4? (a) 1050 (b) 1042 (c) 1044 (d) 1047
10. 10.What is the remainder when 92 is divided by 6? (a) 3 (b) 1 (c) 4 (d) 2
11. 11.What is the units digit of 9^57? (a) 7 (b) 8 (c) 1 (d) 9
12. 12.What is the smallest number you must add to 255 to make it divisible by 10? (a) 4 (b) 7 (c) 5 (d) 6
13. 13.What is the remainder when 445 is divided by 6? (a) 0 (b) 1 (c) 5 (d) 2
14. 14.What is the remainder when 125 is divided by 7? (a) 1 (b) 5 (c) 4 (d) 6
15. 15.Which of these numbers is divisible by 6? (a) 4177 (b) 4182 (c) 4191 (d) 4184
16. 16.What is the smallest number you must add to 174 to make it divisible by 12? (a) 6 (b) 7 (c) 8 (d) 5
17. 17.What is the smallest number you must add to 439 to make it divisible by 7? (a) 5 (b) 2 (c) 3 (d) 1
18. 18.What is the smallest number you must add to 741 to make it divisible by 11? (a) 4 (b) 8 (c) 7 (d) 6
19. 19.What is the remainder when 445 is divided by 12? (a) 11 (b) 0 (c) 1 (d) 2
20. 20.What is the remainder when 842 is divided by 12? (a) 10 (b) 3 (c) 2 (d) 1
21. 21.What is the units digit of 4^42? (a) 4 (b) 5 (c) 6 (d) 7
22. 22.Which of these numbers is divisible by 8? (a) 4013 (b) 4003 (c) 4018 (d) 4008
23. 23.Which of these numbers is divisible by 8? (a) 3646 (b) 3651 (c) 3648 (d) 3657
24. 24.What is the smallest number you must add to 142 to make it divisible by 3? (a) 1 (b) 2
25. 25.Which of these numbers is divisible by 3? (a) 2382 (b) 2386 (c) 2383 (d) 2381
26. 26.Which of these numbers is divisible by 11? (a) 612 (b) 603 (c) 605 (d) 618
27. 27.What is the units digit of 7^40? (a) 9 (b) 3 (c) 1 (d) 7
28. 28.What is the units digit of 2^58? (a) 4 (b) 8 (c) 2 (d) 6
29. 29.What is the units digit of 2^49? (a) 6 (b) 4 (c) 2 (d) 8
30. 30.What is the remainder when 471 is divided by 8? (a) 7 (b) 6 (c) 5 (d) 1
31. 31.What is the units digit of 2^19? (a) 6 (b) 8 (c) 4 (d) 2
32. 32.What is the remainder when 422 is divided by 7? (a) 3 (b) 5 (c) 2 (d) 1
33. 33.What is the smallest number you must add to 606 to make it divisible by 9? (a) 3 (b) 7 (c) 5 (d) 6
34. 34.What is the smallest number you must add to 130 to make it divisible by 7? (a) 3 (b) 2 (c) 4 (d) 5
35. 35.What is the smallest number you must add to 162 to make it divisible by 11? (a) 8 (b) 2 (c) 4 (d) 3
36. 36.What is the smallest number you must add to 43 to make it divisible by 4? (a) 1 (b) 2 (c) 3
37. 37.What is the remainder when 621 is divided by 12? (a) 9 (b) 10 (c) 3 (d) 8
38. 38.What is the smallest number you must add to 31 to make it divisible by 3? (a) 1 (b) 2
39. 39.What is the remainder when 605 is divided by 10? (a) 4 (b) 6 (c) 7 (d) 5
40. 40.What is the remainder when 596 is divided by 9? (a) 1 (b) 7 (c) 3 (d) 2

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