Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 147 · 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 9^31? (a) 9 (b) 8 (c) 7 (d) 1
2. 2.Which of these numbers is divisible by 3? (a) 856 (b) 851 (c) 854 (d) 852
3. 3.What is the units digit of 2^41? (a) 4 (b) 6 (c) 2 (d) 8
4. 4.What is the remainder when 757 is divided by 10? (a) 8 (b) 7 (c) 3 (d) 6
5. 5.Which of these numbers is divisible by 11? (a) 5004 (b) 4983 (c) 4989 (d) 4978
6. 6.What is the units digit of 8^35? (a) 6 (b) 4 (c) 2 (d) 8
7. 7.What is the smallest number you must add to 447 to make it divisible by 9? (a) 2 (b) 6 (c) 4 (d) 3
8. 8.What is the smallest number you must add to 544 to make it divisible by 7? (a) 5 (b) 2 (c) 3 (d) 1
9. 9.What is the remainder when 436 is divided by 10? (a) 4 (b) 6 (c) 5 (d) 7
10. 10.Which of these numbers is divisible by 8? (a) 2759 (b) 2750 (c) 2762 (d) 2752
11. 11.What is the smallest number you must add to 113 to make it divisible by 11? (a) 8 (b) 9 (c) 7 (d) 3
12. 12.What is the units digit of 7^33? (a) 9 (b) 1 (c) 7 (d) 3
13. 13.What is the remainder when 259 is divided by 9? (a) 7 (b) 8 (c) 6 (d) 2
14. 14.Which of these numbers is divisible by 3? (a) 2250 (b) 2255 (c) 2248 (d) 2251
15. 15.What is the smallest number you must add to 297 to make it divisible by 7? (a) 4 (b) 6 (c) 5 (d) 3
16. 16.What is the smallest number you must add to 151 to make it divisible by 11? (a) 2 (b) 4 (c) 3 (d) 8
17. 17.What is the units digit of 4^58? (a) 5 (b) 4 (c) 6 (d) 7
18. 18.Which of these numbers is divisible by 8? (a) 867 (b) 849 (c) 856 (d) 861
19. 19.What is the smallest number you must add to 123 to make it divisible by 5? (a) 4 (b) 3 (c) 2 (d) 1
20. 20.What is the units digit of 4^21? (a) 4 (b) 3 (c) 5 (d) 6
21. 21.What is the units digit of 9^43? (a) 1 (b) 9 (c) 7 (d) 8
22. 22.What is the remainder when 570 is divided by 8? (a) 1 (b) 2 (c) 6 (d) 3
23. 23.Which of these numbers is divisible by 3? (a) 769 (b) 768 (c) 766 (d) 773
24. 24.What is the remainder when 805 is divided by 11? (a) 1 (b) 2 (c) 3 (d) 9
25. 25.What is the units digit of 3^39? (a) 7 (b) 3 (c) 1 (d) 9
26. 26.What is the smallest number you must add to 285 to make it divisible by 11? (a) 10 (b) 2 (c) 3 (d) 1
27. 27.What is the units digit of 2^21? (a) 8 (b) 2 (c) 4 (d) 6
28. 28.Which of these numbers is divisible by 8? (a) 5718 (b) 5704 (c) 5705 (d) 5702
29. 29.Which of these numbers is divisible by 9? (a) 5976 (b) 5981 (c) 5971 (d) 5993
30. 30.Which of these numbers is divisible by 3? (a) 1651 (b) 1649 (c) 1654 (d) 1650
31. 31.What is the smallest number you must add to 207 to make it divisible by 4? (a) 3 (b) 2 (c) 1
32. 32.What is the remainder when 61 is divided by 3? (a) 1 (b) 2 (c) 0
33. 33.What is the remainder when 606 is divided by 9? (a) 3 (b) 2 (c) 4 (d) 6
34. 34.What is the remainder when 332 is divided by 7? (a) 2 (b) 4 (c) 5 (d) 3
35. 35.Which of these numbers is divisible by 9? (a) 1669 (b) 1659 (c) 1656 (d) 1654
36. 36.What is the smallest number you must add to 716 to make it divisible by 12? (a) 8 (b) 4 (c) 5 (d) 3
37. 37.What is the remainder when 124 is divided by 10? (a) 6 (b) 4 (c) 5 (d) 3
38. 38.Which of these numbers is divisible by 11? (a) 9042 (b) 9046 (c) 9054 (d) 9040
39. 39.Which of these numbers is divisible by 9? (a) 4500 (b) 4512 (c) 4497 (d) 4507
40. 40.Which of these numbers is divisible by 3? (a) 980 (b) 983 (c) 981 (d) 985

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