Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 657 · 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 smallest number you must add to 205 to make it divisible by 4? (a) 1 (b) 3 (c) 2
2. 2.Which of these numbers is divisible by 11? (a) 4539 (b) 4549 (c) 4543 (d) 4558
3. 3.Which of these numbers is divisible by 4? (a) 1086 (b) 1084 (c) 1081 (d) 1091
4. 4.What is the smallest number you must add to 368 to make it divisible by 5? (a) 2 (b) 1 (c) 3 (d) 4
5. 5.Which of these numbers is divisible by 6? (a) 659 (b) 648 (c) 644 (d) 651
6. 6.Which of these numbers is divisible by 4? (a) 682 (b) 676 (c) 678 (d) 673
7. 7.What is the units digit of 9^45? (a) 1 (b) 8 (c) 7 (d) 9
8. 8.Which of these numbers is divisible by 6? (a) 676 (b) 688 (c) 679 (d) 678
9. 9.Which of these numbers is divisible by 6? (a) 3040 (b) 3043 (c) 3042 (d) 3049
10. 10.What is the units digit of 2^32? (a) 8 (b) 2 (c) 4 (d) 6
11. 11.What is the remainder when 394 is divided by 9? (a) 7 (b) 2 (c) 6 (d) 8
12. 12.Which of these numbers is divisible by 8? (a) 4193 (b) 4189 (c) 4177 (d) 4184
13. 13.Which of these numbers is divisible by 6? (a) 3188 (b) 3183 (c) 3186 (d) 3196
14. 14.What is the remainder when 170 is divided by 7? (a) 5 (b) 2 (c) 1 (d) 3
15. 15.Which of these numbers is divisible by 11? (a) 8512 (b) 8503 (c) 8501 (d) 8521
16. 16.What is the smallest number you must add to 157 to make it divisible by 3? (a) 2 (b) 1
17. 17.What is the remainder when 133 is divided by 3? (a) 1 (b) 2 (c) 0
18. 18.Which of these numbers is divisible by 6? (a) 4668 (b) 4671 (c) 4676 (d) 4663
19. 19.Which of these numbers is divisible by 4? (a) 1641 (b) 1644 (c) 1647 (d) 1649
20. 20.What is the remainder when 55 is divided by 3? (a) 1 (b) 2 (c) 0
21. 21.What is the smallest number you must add to 857 to make it divisible by 11? (a) 3 (b) 1 (c) 10 (d) 2
22. 22.What is the smallest number you must add to 126 to make it divisible by 4? (a) 3 (b) 1 (c) 2
23. 23.What is the units digit of 2^29? (a) 2 (b) 8 (c) 6 (d) 4
24. 24.What is the units digit of 7^12? (a) 3 (b) 9 (c) 7 (d) 1
25. 25.What is the smallest number you must add to 144 to make it divisible by 7? (a) 4 (b) 2 (c) 3 (d) 5
26. 26.What is the remainder when 227 is divided by 8? (a) 2 (b) 3 (c) 5 (d) 4
27. 27.Which of these numbers is divisible by 6? (a) 3679 (b) 3678 (c) 3673 (d) 3688
28. 28.Which of these numbers is divisible by 4? (a) 1823 (b) 1816 (c) 1818 (d) 1814
29. 29.What is the remainder when 188 is divided by 10? (a) 9 (b) 8 (c) 2 (d) 7
30. 30.What is the units digit of 4^29? (a) 6 (b) 3 (c) 4 (d) 5
31. 31.Which of these numbers is divisible by 9? (a) 2584 (b) 2574 (c) 2579 (d) 2566
32. 32.Which of these numbers is divisible by 6? (a) 2005 (b) 1994 (c) 1998 (d) 2000
33. 33.Which of these numbers is divisible by 9? (a) 1104 (b) 1107 (c) 1117 (d) 1108
34. 34.What is the remainder when 65 is divided by 6? (a) 5 (b) 3 (c) 4 (d) 1
35. 35.What is the units digit of 3^28? (a) 1 (b) 7 (c) 9 (d) 3
36. 36.What is the units digit of 9^34? (a) 2 (b) 1 (c) 0 (d) 9
37. 37.Which of these numbers is divisible by 3? (a) 2414 (b) 2415 (c) 2417 (d) 2419
38. 38.Which of these numbers is divisible by 6? (a) 5136 (b) 5144 (c) 5140 (d) 5134
39. 39.What is the units digit of 3^12? (a) 3 (b) 1 (c) 7 (d) 9
40. 40.What is the remainder when 242 is divided by 3? (a) 2 (b) 0 (c) 1

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