Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 71 · 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 577 is divided by 10? (a) 8 (b) 7 (c) 3 (d) 6
2. 2.What is the units digit of 9^57? (a) 7 (b) 1 (c) 8 (d) 9
3. 3.Which of these numbers is divisible by 11? (a) 1835 (b) 1845 (c) 1826 (d) 1825
4. 4.What is the units digit of 7^37? (a) 1 (b) 9 (c) 3 (d) 7
5. 5.What is the remainder when 155 is divided by 3? (a) 2 (b) 1 (c) 0
6. 6.What is the units digit of 2^45? (a) 8 (b) 2 (c) 6 (d) 4
7. 7.What is the remainder when 83 is divided by 4? (a) 0 (b) 2 (c) 3 (d) 1
8. 8.Which of these numbers is divisible by 11? (a) 5493 (b) 5489 (c) 5486 (d) 5501
9. 9.What is the smallest number you must add to 47 to make it divisible by 3? (a) 2 (b) 1
10. 10.What is the units digit of 2^21? (a) 8 (b) 6 (c) 2 (d) 4
11. 11.What is the remainder when 317 is divided by 9? (a) 7 (b) 1 (c) 3 (d) 2
12. 12.Which of these numbers is divisible by 9? (a) 2513 (b) 2511 (c) 2521 (d) 2504
13. 13.What is the units digit of 2^53? (a) 8 (b) 6 (c) 4 (d) 2
14. 14.What is the units digit of 2^38? (a) 2 (b) 4 (c) 8 (d) 6
15. 15.What is the smallest number you must add to 92 to make it divisible by 3? (a) 1 (b) 2
16. 16.What is the smallest number you must add to 164 to make it divisible by 3? (a) 2 (b) 1
17. 17.What is the remainder when 388 is divided by 7? (a) 4 (b) 2 (c) 5 (d) 3
18. 18.Which of these numbers is divisible by 6? (a) 2283 (b) 2274 (c) 2279 (d) 2270
19. 19.Which of these numbers is divisible by 11? (a) 9794 (b) 9806 (c) 9814 (d) 9801
20. 20.Which of these numbers is divisible by 9? (a) 863 (b) 841 (c) 851 (d) 846
21. 21.What is the smallest number you must add to 402 to make it divisible by 8? (a) 7 (b) 5 (c) 2 (d) 6
22. 22.What is the remainder when 332 is divided by 10? (a) 3 (b) 2 (c) 8 (d) 1
23. 23.What is the smallest number you must add to 145 to make it divisible by 7? (a) 3 (b) 1 (c) 2 (d) 5
24. 24.What is the units digit of 2^42? (a) 8 (b) 6 (c) 2 (d) 4
25. 25.What is the remainder when 82 is divided by 8? (a) 6 (b) 3 (c) 2 (d) 1
26. 26.What is the remainder when 363 is divided by 7? (a) 5 (b) 4 (c) 6 (d) 1
27. 27.What is the remainder when 383 is divided by 11? (a) 2 (b) 10 (c) 9 (d) 8
28. 28.Which of these numbers is divisible by 3? (a) 767 (b) 770 (c) 765 (d) 764
29. 29.What is the units digit of 2^52? (a) 6 (b) 2 (c) 8 (d) 4
30. 30.What is the remainder when 794 is divided by 11? (a) 9 (b) 1 (c) 2 (d) 3
31. 31.Which of these numbers is divisible by 3? (a) 2654 (b) 2650 (c) 2652 (d) 2657
32. 32.What is the units digit of 4^22? (a) 5 (b) 7 (c) 6 (d) 4
33. 33.What is the smallest number you must add to 68 to make it divisible by 3? (a) 1 (b) 2
34. 34.What is the units digit of 9^28? (a) 1 (b) 0 (c) 2 (d) 9
35. 35.What is the units digit of 8^45? (a) 6 (b) 2 (c) 4 (d) 8
36. 36.What is the units digit of 2^30? (a) 8 (b) 4 (c) 2 (d) 6
37. 37.What is the units digit of 8^38? (a) 8 (b) 2 (c) 4 (d) 6
38. 38.What is the units digit of 3^34? (a) 3 (b) 7 (c) 9 (d) 1
39. 39.Which of these numbers is divisible by 3? (a) 1072 (b) 1075 (c) 1069 (d) 1071
40. 40.What is the units digit of 3^54? (a) 7 (b) 9 (c) 3 (d) 1

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