Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 989 · 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 2^53? (a) 4 (b) 2 (c) 6 (d) 8
2. 2.Which of these numbers is divisible by 3? (a) 889 (b) 888 (c) 886 (d) 892
3. 3.What is the remainder when 150 is divided by 9? (a) 5 (b) 7 (c) 3 (d) 6
4. 4.Which of these numbers is divisible by 3? (a) 233 (b) 230 (c) 231 (d) 236
5. 5.What is the units digit of 4^39? (a) 5 (b) 6 (c) 4 (d) 3
6. 6.Which of these numbers is divisible by 9? (a) 3915 (b) 3913 (c) 3926 (d) 3919
7. 7.What is the smallest number you must add to 441 to make it divisible by 12? (a) 2 (b) 4 (c) 3 (d) 9
8. 8.What is the units digit of 3^56? (a) 3 (b) 1 (c) 7 (d) 9
9. 9.What is the units digit of 9^37? (a) 8 (b) 1 (c) 7 (d) 9
10. 10.What is the smallest number you must add to 426 to make it divisible by 9? (a) 3 (b) 5 (c) 7 (d) 6
11. 11.What is the units digit of 2^19? (a) 6 (b) 4 (c) 8 (d) 2
12. 12.Which of these numbers is divisible by 4? (a) 2799 (b) 2801 (c) 2796 (d) 2793
13. 13.What is the units digit of 8^49? (a) 8 (b) 2 (c) 6 (d) 4
14. 14.What is the remainder when 418 is divided by 10? (a) 9 (b) 7 (c) 8 (d) 2
15. 15.What is the remainder when 177 is divided by 6? (a) 5 (b) 4 (c) 3 (d) 2
16. 16.Which of these numbers is divisible by 8? (a) 2872 (b) 2887 (c) 2876 (d) 2871
17. 17.What is the smallest number you must add to 309 to make it divisible by 5? (a) 2 (b) 3 (c) 1 (d) 4
18. 18.What is the units digit of 8^45? (a) 6 (b) 8 (c) 4 (d) 2
19. 19.Which of these numbers is divisible by 8? (a) 6959 (b) 6975 (c) 6960 (d) 6965
20. 20.What is the units digit of 3^50? (a) 3 (b) 9 (c) 1 (d) 7
21. 21.What is the units digit of 8^27? (a) 4 (b) 8 (c) 2 (d) 6
22. 22.What is the units digit of 3^47? (a) 3 (b) 9 (c) 7 (d) 1
23. 23.What is the smallest number you must add to 58 to make it divisible by 3? (a) 2 (b) 1
24. 24.What is the units digit of 9^19? (a) 1 (b) 9 (c) 8 (d) 7
25. 25.Which of these numbers is divisible by 8? (a) 5673 (b) 5686 (c) 5680 (d) 5692
26. 26.What is the units digit of 3^60? (a) 9 (b) 3 (c) 7 (d) 1
27. 27.What is the smallest number you must add to 186 to make it divisible by 10? (a) 5 (b) 3 (c) 4 (d) 6
28. 28.Which of these numbers is divisible by 4? (a) 2270 (b) 2265 (c) 2263 (d) 2264
29. 29.What is the units digit of 4^53? (a) 6 (b) 5 (c) 3 (d) 4
30. 30.What is the units digit of 4^49? (a) 3 (b) 5 (c) 4 (d) 6
31. 31.What is the remainder when 128 is divided by 3? (a) 1 (b) 2 (c) 0
32. 32.Which of these numbers is divisible by 4? (a) 3432 (b) 3437 (c) 3434 (d) 3431
33. 33.What is the remainder when 420 is divided by 11? (a) 1 (b) 9 (c) 2 (d) 3
34. 34.What is the units digit of 7^33? (a) 9 (b) 7 (c) 3 (d) 1
35. 35.What is the remainder when 141 is divided by 5? (a) 0 (b) 4 (c) 1 (d) 2
36. 36.Which of these numbers is divisible by 3? (a) 1803 (b) 1802 (c) 1805 (d) 1808
37. 37.Which of these numbers is divisible by 3? (a) 885 (b) 883 (c) 889 (d) 886
38. 38.What is the units digit of 4^55? (a) 4 (b) 3 (c) 5 (d) 6
39. 39.What is the units digit of 2^41? (a) 4 (b) 2 (c) 6 (d) 8
40. 40.What is the remainder when 169 is divided by 8? (a) 2 (b) 0 (c) 7 (d) 1

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