Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 840 · 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 9^13? (a) 8 (b) 9 (c) 7 (d) 1
2. 2.What is the remainder when 394 is divided by 9? (a) 2 (b) 7 (c) 8 (d) 6
3. 3.What is the smallest number you must add to 387 to make it divisible by 5? (a) 1 (b) 2 (c) 4 (d) 3
4. 4.What is the units digit of 9^12? (a) 9 (b) 2 (c) 0 (d) 1
5. 5.Which of these numbers is divisible by 3? (a) 426 (b) 430 (c) 427 (d) 425
6. 6.Which of these numbers is divisible by 11? (a) 2915 (b) 2930 (c) 2919 (d) 2913
7. 7.What is the smallest number you must add to 430 to make it divisible by 11? (a) 10 (b) 8 (c) 9 (d) 1
8. 8.What is the units digit of 2^53? (a) 4 (b) 6 (c) 2 (d) 8
9. 9.What is the smallest number you must add to 466 to make it divisible by 10? (a) 5 (b) 4 (c) 6 (d) 3
10. 10.What is the smallest number you must add to 243 to make it divisible by 11? (a) 1 (b) 10 (c) 8 (d) 9
11. 11.What is the units digit of 8^23? (a) 6 (b) 8 (c) 4 (d) 2
12. 12.Which of these numbers is divisible by 8? (a) 6519 (b) 6504 (c) 6499 (d) 6505
13. 13.What is the smallest number you must add to 191 to make it divisible by 3? (a) 2 (b) 1
14. 14.Which of these numbers is divisible by 4? (a) 2045 (b) 2040 (c) 2043 (d) 2037
15. 15.What is the units digit of 9^54? (a) 9 (b) 0 (c) 1 (d) 2
16. 16.What is the remainder when 419 is divided by 7? (a) 5 (b) 6 (c) 4 (d) 1
17. 17.What is the remainder when 437 is divided by 9? (a) 4 (b) 7 (c) 5 (d) 6
18. 18.Which of these numbers is divisible by 4? (a) 571 (b) 573 (c) 568 (d) 565
19. 19.What is the remainder when 138 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
20. 20.Which of these numbers is divisible by 9? (a) 5130 (b) 5145 (c) 5123 (d) 5132
21. 21.What is the units digit of 3^40? (a) 1 (b) 3 (c) 9 (d) 7
22. 22.What is the smallest number you must add to 247 to make it divisible by 5? (a) 3 (b) 4 (c) 1 (d) 2
23. 23.What is the units digit of 8^46? (a) 6 (b) 2 (c) 8 (d) 4
24. 24.What is the units digit of 3^37? (a) 9 (b) 1 (c) 7 (d) 3
25. 25.What is the units digit of 8^14? (a) 4 (b) 8 (c) 2 (d) 6
26. 26.What is the units digit of 2^42? (a) 8 (b) 2 (c) 6 (d) 4
27. 27.Which of these numbers is divisible by 8? (a) 6059 (b) 6042 (c) 6052 (d) 6048
28. 28.What is the remainder when 300 is divided by 7? (a) 4 (b) 1 (c) 6 (d) 5
29. 29.What is the smallest number you must add to 183 to make it divisible by 4? (a) 2 (b) 1 (c) 3
30. 30.What is the units digit of 8^37? (a) 2 (b) 6 (c) 4 (d) 8
31. 31.What is the smallest number you must add to 474 to make it divisible by 10? (a) 4 (b) 7 (c) 5 (d) 6
32. 32.Which of these numbers is divisible by 9? (a) 6241 (b) 6263 (c) 6251 (d) 6246
33. 33.What is the remainder when 133 is divided by 8? (a) 4 (b) 3 (c) 5 (d) 6
34. 34.What is the remainder when 206 is divided by 5? (a) 4 (b) 1 (c) 0 (d) 2
35. 35.What is the remainder when 203 is divided by 3? (a) 0 (b) 2 (c) 1
36. 36.What is the units digit of 3^29? (a) 7 (b) 9 (c) 3 (d) 1
37. 37.Which of these numbers is divisible by 6? (a) 4991 (b) 4997 (c) 4985 (d) 4986
38. 38.Which of these numbers is divisible by 4? (a) 3112 (b) 3111 (c) 3117 (d) 3114
39. 39.What is the remainder when 460 is divided by 6? (a) 2 (b) 5 (c) 3 (d) 4
40. 40.Which of these numbers is divisible by 4? (a) 678 (b) 676 (c) 673 (d) 683

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