Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 658 · 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 8^12? (a) 4 (b) 8 (c) 6 (d) 2
2. 2.What is the remainder when 531 is divided by 11? (a) 2 (b) 3 (c) 4 (d) 8
3. 3.What is the units digit of 3^38? (a) 9 (b) 1 (c) 7 (d) 3
4. 4.Which of these numbers is divisible by 9? (a) 5105 (b) 5101 (c) 5086 (d) 5094
5. 5.Which of these numbers is divisible by 4? (a) 849 (b) 842 (c) 844 (d) 845
6. 6.What is the smallest number you must add to 326 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
7. 7.What is the units digit of 7^13? (a) 9 (b) 3 (c) 1 (d) 7
8. 8.What is the remainder when 205 is divided by 8? (a) 5 (b) 4 (c) 6 (d) 3
9. 9.Which of these numbers is divisible by 9? (a) 4029 (b) 4039 (c) 4023 (d) 4020
10. 10.Which of these numbers is divisible by 8? (a) 1885 (b) 1872 (c) 1871 (d) 1879
11. 11.What is the units digit of 7^49? (a) 3 (b) 7 (c) 9 (d) 1
12. 12.What is the remainder when 122 is divided by 11? (a) 0 (b) 1 (c) 2 (d) 10
13. 13.Which of these numbers is divisible by 4? (a) 467 (b) 462 (c) 469 (d) 464
14. 14.What is the smallest number you must add to 398 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
15. 15.Which of these numbers is divisible by 9? (a) 2251 (b) 2241 (c) 2249 (d) 2233
16. 16.What is the units digit of 4^21? (a) 6 (b) 3 (c) 4 (d) 5
17. 17.What is the smallest number you must add to 429 to make it divisible by 6? (a) 5 (b) 4 (c) 2 (d) 3
18. 18.What is the smallest number you must add to 307 to make it divisible by 10? (a) 4 (b) 2 (c) 7 (d) 3
19. 19.What is the remainder when 168 is divided by 9? (a) 7 (b) 5 (c) 3 (d) 6
20. 20.Which of these numbers is divisible by 9? (a) 6487 (b) 6502 (c) 6489 (d) 6497
21. 21.What is the units digit of 8^22? (a) 4 (b) 6 (c) 8 (d) 2
22. 22.What is the remainder when 125 is divided by 9? (a) 8 (b) 7 (c) 6 (d) 1
23. 23.Which of these numbers is divisible by 11? (a) 1176 (b) 1194 (c) 1177 (d) 1179
24. 24.What is the smallest number you must add to 95 to make it divisible by 3? (a) 1 (b) 2
25. 25.What is the remainder when 301 is divided by 6? (a) 2 (b) 5 (c) 1 (d) 0
26. 26.Which of these numbers is divisible by 9? (a) 7129 (b) 7126 (c) 7119 (d) 7113
27. 27.What is the remainder when 644 is divided by 12? (a) 7 (b) 9 (c) 4 (d) 8
28. 28.What is the units digit of 8^41? (a) 4 (b) 8 (c) 2 (d) 6
29. 29.What is the units digit of 3^12? (a) 3 (b) 7 (c) 1 (d) 9
30. 30.Which of these numbers is divisible by 3? (a) 2582 (b) 2585 (c) 2578 (d) 2580
31. 31.What is the remainder when 98 is divided by 9? (a) 8 (b) 7 (c) 6 (d) 1
32. 32.What is the remainder when 564 is divided by 11? (a) 4 (b) 8 (c) 2 (d) 3
33. 33.What is the smallest number you must add to 447 to make it divisible by 7? (a) 1 (b) 2 (c) 3 (d) 6
34. 34.What is the remainder when 573 is divided by 9? (a) 3 (b) 5 (c) 7 (d) 6
35. 35.What is the remainder when 365 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 5
36. 36.What is the smallest number you must add to 193 to make it divisible by 4? (a) 3 (b) 1 (c) 2
37. 37.What is the smallest number you must add to 458 to make it divisible by 8? (a) 5 (b) 6 (c) 2 (d) 7
38. 38.What is the units digit of 8^33? (a) 2 (b) 4 (c) 6 (d) 8
39. 39.What is the remainder when 222 is divided by 7? (a) 6 (b) 2 (c) 5 (d) 4
40. 40.What is the remainder when 45 is divided by 4? (a) 3 (b) 1 (c) 2 (d) 0

Free maths olympiad lessons, practice and worksheets at talentjr.in/maths-olympiad