Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 228 · 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 9^51? (a) 1 (b) 8 (c) 9 (d) 7
2. 2.What is the smallest number you must add to 130 to make it divisible by 3? (a) 1 (b) 2
3. 3.Which of these numbers is divisible by 3? (a) 1674 (b) 1679 (c) 1673 (d) 1675
4. 4.What is the smallest number you must add to 97 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
5. 5.What is the units digit of 9^47? (a) 1 (b) 8 (c) 7 (d) 9
6. 6.What is the smallest number you must add to 263 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
7. 7.What is the units digit of 8^13? (a) 4 (b) 6 (c) 8 (d) 2
8. 8.What is the units digit of 4^35? (a) 3 (b) 4 (c) 5 (d) 6
9. 9.What is the units digit of 7^48? (a) 7 (b) 9 (c) 1 (d) 3
10. 10.What is the remainder when 217 is divided by 8? (a) 7 (b) 1 (c) 2 (d) 0
11. 11.What is the units digit of 4^26? (a) 5 (b) 4 (c) 6 (d) 7
12. 12.What is the units digit of 4^56? (a) 7 (b) 6 (c) 5 (d) 4
13. 13.What is the units digit of 2^42? (a) 4 (b) 6 (c) 8 (d) 2
14. 14.Which of these numbers is divisible by 9? (a) 3240 (b) 3233 (c) 3244 (d) 3254
15. 15.What is the smallest number you must add to 366 to make it divisible by 7? (a) 2 (b) 4 (c) 5 (d) 6
16. 16.What is the units digit of 7^56? (a) 9 (b) 3 (c) 7 (d) 1
17. 17.What is the smallest number you must add to 395 to make it divisible by 12? (a) 11 (b) 1 (c) 3 (d) 2
18. 18.What is the smallest number you must add to 389 to make it divisible by 5? (a) 4 (b) 2 (c) 1 (d) 3
19. 19.Which of these numbers is divisible by 3? (a) 1187 (b) 1189 (c) 1184 (d) 1185
20. 20.What is the remainder when 88 is divided by 5? (a) 2 (b) 1 (c) 4 (d) 3
21. 21.Which of these numbers is divisible by 3? (a) 1667 (b) 1670 (c) 1664 (d) 1665
22. 22.What is the units digit of 8^22? (a) 6 (b) 2 (c) 8 (d) 4
23. 23.Which of these numbers is divisible by 3? (a) 513 (b) 515 (c) 511 (d) 518
24. 24.What is the smallest number you must add to 71 to make it divisible by 6? (a) 5 (b) 3 (c) 1 (d) 2
25. 25.What is the remainder when 324 is divided by 8? (a) 4 (b) 6 (c) 5 (d) 3
26. 26.What is the remainder when 170 is divided by 6? (a) 4 (b) 2 (c) 1 (d) 3
27. 27.What is the units digit of 8^17? (a) 2 (b) 4 (c) 6 (d) 8
28. 28.What is the smallest number you must add to 244 to make it divisible by 5? (a) 3 (b) 2 (c) 4 (d) 1
29. 29.What is the smallest number you must add to 314 to make it divisible by 10? (a) 5 (b) 4 (c) 6 (d) 7
30. 30.What is the remainder when 152 is divided by 3? (a) 1 (b) 2 (c) 0
31. 31.What is the units digit of 7^42? (a) 1 (b) 7 (c) 3 (d) 9
32. 32.What is the remainder when 162 is divided by 4? (a) 2 (b) 3 (c) 1 (d) 0
33. 33.What is the remainder when 403 is divided by 12? (a) 6 (b) 5 (c) 8 (d) 7
34. 34.What is the units digit of 7^11? (a) 3 (b) 9 (c) 7 (d) 1
35. 35.What is the smallest number you must add to 275 to make it divisible by 12? (a) 2 (b) 1 (c) 11 (d) 3
36. 36.What is the smallest number you must add to 103 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
37. 37.Which of these numbers is divisible by 11? (a) 7627 (b) 7620 (c) 7607 (d) 7612
38. 38.What is the units digit of 3^59? (a) 9 (b) 1 (c) 7 (d) 3
39. 39.What is the smallest number you must add to 115 to make it divisible by 4? (a) 3 (b) 1 (c) 2
40. 40.Which of these numbers is divisible by 6? (a) 486 (b) 494 (c) 484 (d) 487

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