Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 107 · 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 3^41? (a) 1 (b) 7 (c) 3 (d) 9
2. 2.What is the units digit of 3^40? (a) 1 (b) 3 (c) 9 (d) 7
3. 3.What is the remainder when 347 is divided by 12? (a) 11 (b) 10 (c) 9 (d) 1
4. 4.What is the remainder when 140 is divided by 8? (a) 4 (b) 3 (c) 6 (d) 5
5. 5.What is the smallest number you must add to 258 to make it divisible by 10? (a) 3 (b) 2 (c) 1 (d) 8
6. 6.What is the remainder when 269 is divided by 5? (a) 1 (b) 4 (c) 3 (d) 2
7. 7.What is the units digit of 2^40? (a) 4 (b) 2 (c) 8 (d) 6
8. 8.What is the smallest number you must add to 53 to make it divisible by 3? (a) 2 (b) 1
9. 9.Which of these numbers is divisible by 6? (a) 1516 (b) 1523 (c) 1526 (d) 1518
10. 10.What is the remainder when 639 is divided by 8? (a) 6 (b) 7 (c) 5 (d) 1
11. 11.What is the remainder when 626 is divided by 8? (a) 6 (b) 3 (c) 1 (d) 2
12. 12.What is the remainder when 156 is divided by 11? (a) 1 (b) 2 (c) 3 (d) 9
13. 13.What is the remainder when 519 is divided by 9? (a) 7 (b) 6 (c) 5 (d) 3
14. 14.What is the units digit of 4^40? (a) 5 (b) 6 (c) 7 (d) 4
15. 15.What is the smallest number you must add to 157 to make it divisible by 11? (a) 3 (b) 7 (c) 9 (d) 8
16. 16.What is the units digit of 4^54? (a) 6 (b) 4 (c) 7 (d) 5
17. 17.What is the units digit of 9^40? (a) 9 (b) 2 (c) 1 (d) 0
18. 18.Which of these numbers is divisible by 6? (a) 1640 (b) 1632 (c) 1637 (d) 1627
19. 19.What is the remainder when 158 is divided by 6? (a) 4 (b) 1 (c) 3 (d) 2
20. 20.Which of these numbers is divisible by 8? (a) 1088 (b) 1084 (c) 1094 (d) 1102
21. 21.What is the units digit of 9^50? (a) 2 (b) 9 (c) 0 (d) 1
22. 22.Which of these numbers is divisible by 6? (a) 2665 (b) 2672 (c) 2662 (d) 2664
23. 23.What is the units digit of 2^16? (a) 8 (b) 6 (c) 2 (d) 4
24. 24.Which of these numbers is divisible by 4? (a) 1333 (b) 1336 (c) 1341 (d) 1337
25. 25.What is the smallest number you must add to 136 to make it divisible by 5? (a) 2 (b) 1 (c) 4 (d) 3
26. 26.What is the remainder when 145 is divided by 12? (a) 0 (b) 2 (c) 11 (d) 1
27. 27.What is the units digit of 3^39? (a) 3 (b) 7 (c) 1 (d) 9
28. 28.What is the remainder when 394 is divided by 9? (a) 2 (b) 7 (c) 6 (d) 8
29. 29.Which of these numbers is divisible by 6? (a) 2819 (b) 2809 (c) 2808 (d) 2806
30. 30.What is the remainder when 125 is divided by 3? (a) 1 (b) 2 (c) 0
31. 31.What is the units digit of 3^22? (a) 3 (b) 7 (c) 9 (d) 1
32. 32.What is the units digit of 7^30? (a) 7 (b) 1 (c) 9 (d) 3
33. 33.What is the remainder when 301 is divided by 11? (a) 5 (b) 7 (c) 4 (d) 3
34. 34.What is the units digit of 2^20? (a) 2 (b) 4 (c) 8 (d) 6
35. 35.What is the units digit of 8^39? (a) 4 (b) 2 (c) 6 (d) 8
36. 36.What is the units digit of 4^41? (a) 3 (b) 4 (c) 5 (d) 6
37. 37.What is the remainder when 421 is divided by 6? (a) 5 (b) 2 (c) 0 (d) 1
38. 38.Which of these numbers is divisible by 3? (a) 1483 (b) 1481 (c) 1482 (d) 1487
39. 39.Which of these numbers is divisible by 6? (a) 5138 (b) 5129 (c) 5132 (d) 5130
40. 40.Which of these numbers is divisible by 4? (a) 1096 (b) 1099 (c) 1094 (d) 1101

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