Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 177 · 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 9^47? (a) 8 (b) 7 (c) 1 (d) 9
2. 2.What is the units digit of 9^45? (a) 9 (b) 7 (c) 8 (d) 1
3. 3.What is the remainder when 569 is divided by 8? (a) 0 (b) 7 (c) 2 (d) 1
4. 4.What is the smallest number you must add to 370 to make it divisible by 11? (a) 4 (b) 7 (c) 5 (d) 3
5. 5.What is the remainder when 129 is divided by 4? (a) 2 (b) 1 (c) 0 (d) 3
6. 6.What is the units digit of 3^35? (a) 9 (b) 3 (c) 1 (d) 7
7. 7.What is the remainder when 589 is divided by 8? (a) 3 (b) 4 (c) 5 (d) 6
8. 8.What is the remainder when 42 is divided by 4? (a) 0 (b) 1 (c) 3 (d) 2
9. 9.What is the smallest number you must add to 366 to make it divisible by 5? (a) 1 (b) 4 (c) 2 (d) 3
10. 10.What is the smallest number you must add to 214 to make it divisible by 3? (a) 1 (b) 2
11. 11.What is the units digit of 8^15? (a) 8 (b) 4 (c) 2 (d) 6
12. 12.What is the remainder when 211 is divided by 10? (a) 2 (b) 0 (c) 9 (d) 1
13. 13.Which of these numbers is divisible by 11? (a) 6776 (b) 6766 (c) 6789 (d) 6777
14. 14.What is the smallest number you must add to 416 to make it divisible by 7? (a) 3 (b) 5 (c) 4 (d) 6
15. 15.What is the smallest number you must add to 176 to make it divisible by 6? (a) 4 (b) 5 (c) 2 (d) 3
16. 16.Which of these numbers is divisible by 4? (a) 2500 (b) 2507 (c) 2497 (d) 2502
17. 17.What is the units digit of 8^22? (a) 6 (b) 8 (c) 4 (d) 2
18. 18.Which of these numbers is divisible by 6? (a) 918 (b) 929 (c) 919 (d) 913
19. 19.What is the remainder when 58 is divided by 5? (a) 4 (b) 2 (c) 3 (d) 1
20. 20.What is the units digit of 2^45? (a) 8 (b) 6 (c) 4 (d) 2
21. 21.What is the units digit of 8^58? (a) 2 (b) 8 (c) 4 (d) 6
22. 22.Which of these numbers is divisible by 8? (a) 750 (b) 752 (c) 761 (d) 756
23. 23.What is the remainder when 98 is divided by 6? (a) 3 (b) 4 (c) 2 (d) 1
24. 24.What is the remainder when 64 is divided by 5? (a) 4 (b) 3 (c) 1 (d) 2
25. 25.What is the remainder when 259 is divided by 4? (a) 1 (b) 0 (c) 2 (d) 3
26. 26.What is the remainder when 133 is divided by 4? (a) 1 (b) 2 (c) 3 (d) 0
27. 27.What is the smallest number you must add to 111 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
28. 28.What is the smallest number you must add to 113 to make it divisible by 4? (a) 3 (b) 1 (c) 2
29. 29.What is the units digit of 2^14? (a) 6 (b) 8 (c) 2 (d) 4
30. 30.What is the smallest number you must add to 91 to make it divisible by 9? (a) 1 (b) 8 (c) 7 (d) 6
31. 31.What is the remainder when 628 is divided by 10? (a) 7 (b) 9 (c) 2 (d) 8
32. 32.What is the smallest number you must add to 77 to make it divisible by 3? (a) 2 (b) 1
33. 33.What is the units digit of 9^16? (a) 0 (b) 1 (c) 2 (d) 9
34. 34.What is the smallest number you must add to 262 to make it divisible by 9? (a) 7 (b) 8 (c) 6 (d) 1
35. 35.What is the units digit of 8^10? (a) 8 (b) 2 (c) 6 (d) 4
36. 36.What is the smallest number you must add to 73 to make it divisible by 4? (a) 2 (b) 3 (c) 1
37. 37.What is the remainder when 330 is divided by 12? (a) 5 (b) 7 (c) 8 (d) 6
38. 38.What is the remainder when 373 is divided by 9? (a) 3 (b) 4 (c) 5 (d) 6
39. 39.What is the smallest number you must add to 55 to make it divisible by 3? (a) 2 (b) 1
40. 40.What is the remainder when 352 is divided by 12? (a) 8 (b) 4 (c) 3 (d) 5

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