Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 145 · 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 531 to make it divisible by 7? (a) 2 (b) 6 (c) 3 (d) 1
2. 2.Which of these numbers is divisible by 4? (a) 2803 (b) 2796 (c) 2798 (d) 2795
3. 3.Which of these numbers is divisible by 6? (a) 2144 (b) 2136 (c) 2137 (d) 2133
4. 4.What is the smallest number you must add to 570 to make it divisible by 12? (a) 6 (b) 5 (c) 8 (d) 7
5. 5.Which of these numbers is divisible by 6? (a) 599 (b) 587 (c) 588 (d) 592
6. 6.What is the remainder when 217 is divided by 8? (a) 2 (b) 1 (c) 0 (d) 7
7. 7.Which of these numbers is divisible by 9? (a) 1727 (b) 1719 (c) 1736 (d) 1712
8. 8.Which of these numbers is divisible by 6? (a) 379 (b) 370 (c) 373 (d) 372
9. 9.What is the smallest number you must add to 289 to make it divisible by 6? (a) 1 (b) 5 (c) 4 (d) 3
10. 10.What is the remainder when 122 is divided by 9? (a) 4 (b) 5 (c) 6 (d) 7
11. 11.What is the remainder when 87 is divided by 7? (a) 2 (b) 5 (c) 4 (d) 3
12. 12.Which of these numbers is divisible by 8? (a) 1650 (b) 1656 (c) 1666 (d) 1662
13. 13.Which of these numbers is divisible by 3? (a) 808 (b) 811 (c) 805 (d) 807
14. 14.What is the remainder when 627 is divided by 8? (a) 4 (b) 3 (c) 2 (d) 5
15. 15.What is the remainder when 227 is divided by 8? (a) 5 (b) 4 (c) 2 (d) 3
16. 16.What is the smallest number you must add to 165 to make it divisible by 6? (a) 3 (b) 5 (c) 2 (d) 4
17. 17.Which of these numbers is divisible by 9? (a) 3304 (b) 3320 (c) 3303 (d) 3295
18. 18.What is the smallest number you must add to 115 to make it divisible by 3? (a) 2 (b) 1
19. 19.What is the units digit of 2^24? (a) 8 (b) 4 (c) 6 (d) 2
20. 20.What is the units digit of 3^11? (a) 9 (b) 7 (c) 3 (d) 1
21. 21.What is the smallest number you must add to 789 to make it divisible by 11? (a) 2 (b) 8 (c) 3 (d) 4
22. 22.What is the remainder when 476 is divided by 10? (a) 4 (b) 5 (c) 7 (d) 6
23. 23.What is the remainder when 167 is divided by 7? (a) 1 (b) 4 (c) 5 (d) 6
24. 24.What is the smallest number you must add to 138 to make it divisible by 4? (a) 2 (b) 1 (c) 3
25. 25.What is the units digit of 3^57? (a) 3 (b) 9 (c) 7 (d) 1
26. 26.Which of these numbers is divisible by 4? (a) 1337 (b) 1341 (c) 1340 (d) 1346
27. 27.What is the remainder when 203 is divided by 11? (a) 4 (b) 5 (c) 6 (d) 7
28. 28.What is the units digit of 2^45? (a) 4 (b) 6 (c) 2 (d) 8
29. 29.What is the smallest number you must add to 805 to make it divisible by 10? (a) 6 (b) 7 (c) 4 (d) 5
30. 30.What is the units digit of 3^26? (a) 9 (b) 7 (c) 1 (d) 3
31. 31.Which of these numbers is divisible by 8? (a) 2487 (b) 2472 (c) 2469 (d) 2474
32. 32.Which of these numbers is divisible by 3? (a) 651 (b) 650 (c) 656 (d) 652
33. 33.What is the remainder when 396 is divided by 7? (a) 4 (b) 6 (c) 3 (d) 5
34. 34.Which of these numbers is divisible by 4? (a) 2789 (b) 2785 (c) 2794 (d) 2788
35. 35.What is the remainder when 522 is divided by 12? (a) 7 (b) 6 (c) 5 (d) 8
36. 36.Which of these numbers is divisible by 9? (a) 6573 (b) 6570 (c) 6586 (d) 6566
37. 37.What is the smallest number you must add to 124 to make it divisible by 3? (a) 1 (b) 2
38. 38.Which of these numbers is divisible by 9? (a) 2834 (b) 2824 (c) 2836 (d) 2826
39. 39.What is the remainder when 147 is divided by 5? (a) 3 (b) 2 (c) 1 (d) 4
40. 40.Which of these numbers is divisible by 6? (a) 4881 (b) 4877 (c) 4888 (d) 4878

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