Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 901 · 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 3^21? (a) 1 (b) 7 (c) 3 (d) 9
2. 2.Which of these numbers is divisible by 11? (a) 9362 (b) 9350 (c) 9348 (d) 9356
3. 3.What is the remainder when 215 is divided by 7? (a) 2 (b) 6 (c) 5 (d) 4
4. 4.What is the remainder when 614 is divided by 8? (a) 2 (b) 5 (c) 7 (d) 6
5. 5.What is the units digit of 4^43? (a) 3 (b) 4 (c) 5 (d) 6
6. 6.What is the smallest number you must add to 293 to make it divisible by 7? (a) 6 (b) 2 (c) 3 (d) 1
7. 7.What is the smallest number you must add to 54 to make it divisible by 4? (a) 1 (b) 3 (c) 2
8. 8.Which of these numbers is divisible by 6? (a) 771 (b) 767 (c) 762 (d) 761
9. 9.What is the remainder when 383 is divided by 5? (a) 2 (b) 4 (c) 1 (d) 3
10. 10.Which of these numbers is divisible by 11? (a) 6668 (b) 6655 (c) 6650 (d) 6656
11. 11.What is the units digit of 9^38? (a) 1 (b) 2 (c) 0 (d) 9
12. 12.Which of these numbers is divisible by 3? (a) 823 (b) 825 (c) 827 (d) 829
13. 13.What is the remainder when 322 is divided by 9? (a) 6 (b) 2 (c) 8 (d) 7
14. 14.What is the remainder when 184 is divided by 7? (a) 1 (b) 5 (c) 2 (d) 3
15. 15.What is the units digit of 9^48? (a) 2 (b) 0 (c) 9 (d) 1
16. 16.What is the units digit of 4^22? (a) 4 (b) 5 (c) 6 (d) 7
17. 17.What is the units digit of 7^49? (a) 1 (b) 7 (c) 3 (d) 9
18. 18.What is the units digit of 9^45? (a) 1 (b) 8 (c) 7 (d) 9
19. 19.What is the remainder when 544 is divided by 12? (a) 4 (b) 8 (c) 3 (d) 5
20. 20.What is the smallest number you must add to 378 to make it divisible by 8? (a) 7 (b) 6 (c) 5 (d) 2
21. 21.Which of these numbers is divisible by 9? (a) 846 (b) 857 (c) 843 (d) 854
22. 22.What is the remainder when 458 is divided by 6? (a) 1 (b) 4 (c) 2 (d) 3
23. 23.What is the remainder when 299 is divided by 7? (a) 4 (b) 6 (c) 5 (d) 2
24. 24.What is the smallest number you must add to 223 to make it divisible by 5? (a) 3 (b) 1 (c) 2 (d) 4
25. 25.What is the smallest number you must add to 265 to make it divisible by 7? (a) 3 (b) 2 (c) 6 (d) 1
26. 26.What is the units digit of 4^50? (a) 6 (b) 7 (c) 4 (d) 5
27. 27.Which of these numbers is divisible by 9? (a) 7093 (b) 7104 (c) 7092 (d) 7091
28. 28.What is the smallest number you must add to 798 to make it divisible by 12? (a) 7 (b) 6 (c) 8 (d) 5
29. 29.What is the remainder when 320 is divided by 9? (a) 4 (b) 7 (c) 5 (d) 6
30. 30.Which of these numbers is divisible by 3? (a) 1825 (b) 1819 (c) 1822 (d) 1821
31. 31.What is the smallest number you must add to 723 to make it divisible by 9? (a) 7 (b) 3 (c) 6 (d) 5
32. 32.What is the units digit of 2^51? (a) 8 (b) 6 (c) 4 (d) 2
33. 33.What is the units digit of 2^17? (a) 8 (b) 6 (c) 2 (d) 4
34. 34.What is the remainder when 93 is divided by 9? (a) 4 (b) 6 (c) 3 (d) 2
35. 35.What is the remainder when 242 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 2
36. 36.Which of these numbers is divisible by 8? (a) 2958 (b) 2947 (c) 2943 (d) 2944
37. 37.What is the units digit of 9^49? (a) 7 (b) 9 (c) 8 (d) 1
38. 38.Which of these numbers is divisible by 3? (a) 1844 (b) 1842 (c) 1846 (d) 1840
39. 39.What is the units digit of 2^29? (a) 6 (b) 2 (c) 8 (d) 4
40. 40.What is the remainder when 472 is divided by 7? (a) 4 (b) 5 (c) 2 (d) 3

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