Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 721 · 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 250 to make it divisible by 4? (a) 1 (b) 3 (c) 2
2. 2.Which of these numbers is divisible by 8? (a) 1256 (b) 1260 (c) 1250 (d) 1268
3. 3.Which of these numbers is divisible by 4? (a) 1651 (b) 1644 (c) 1645 (d) 1643
4. 4.What is the remainder when 451 is divided by 9? (a) 1 (b) 8 (c) 0 (d) 2
5. 5.Which of these numbers is divisible by 4? (a) 803 (b) 809 (c) 805 (d) 804
6. 6.Which of these numbers is divisible by 3? (a) 752 (b) 749 (c) 754 (d) 750
7. 7.What is the remainder when 241 is divided by 5? (a) 1 (b) 0 (c) 2 (d) 4
8. 8.Which of these numbers is divisible by 11? (a) 2731 (b) 2743 (c) 2728 (d) 2724
9. 9.Which of these numbers is divisible by 4? (a) 2050 (b) 2041 (c) 2046 (d) 2044
10. 10.What is the smallest number you must add to 136 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
11. 11.What is the remainder when 285 is divided by 4? (a) 3 (b) 2 (c) 1 (d) 0
12. 12.Which of these numbers is divisible by 6? (a) 4670 (b) 4677 (c) 4682 (d) 4674
13. 13.Which of these numbers is divisible by 9? (a) 6655 (b) 6660 (c) 6677 (d) 6661
14. 14.Which of these numbers is divisible by 3? (a) 2651 (b) 2657 (c) 2652 (d) 2653
15. 15.What is the smallest number you must add to 379 to make it divisible by 9? (a) 6 (b) 7 (c) 1 (d) 8
16. 16.Which of these numbers is divisible by 8? (a) 555 (b) 551 (c) 566 (d) 552
17. 17.What is the units digit of 3^50? (a) 7 (b) 3 (c) 1 (d) 9
18. 18.What is the smallest number you must add to 469 to make it divisible by 8? (a) 3 (b) 2 (c) 4 (d) 5
19. 19.What is the units digit of 2^13? (a) 4 (b) 8 (c) 6 (d) 2
20. 20.What is the remainder when 284 is divided by 6? (a) 2 (b) 3 (c) 4 (d) 1
21. 21.What is the units digit of 9^11? (a) 7 (b) 1 (c) 9 (d) 8
22. 22.Which of these numbers is divisible by 3? (a) 1622 (b) 1624 (c) 1628 (d) 1623
23. 23.Which of these numbers is divisible by 11? (a) 6705 (b) 6698 (c) 6688 (d) 6681
24. 24.What is the remainder when 166 is divided by 8? (a) 2 (b) 7 (c) 5 (d) 6
25. 25.What is the remainder when 130 is divided by 7? (a) 6 (b) 5 (c) 3 (d) 4
26. 26.Which of these numbers is divisible by 3? (a) 1697 (b) 1699 (c) 1698 (d) 1703
27. 27.Which of these numbers is divisible by 3? (a) 2311 (b) 2309 (c) 2315 (d) 2310
28. 28.What is the units digit of 9^25? (a) 9 (b) 1 (c) 8 (d) 7
29. 29.What is the remainder when 474 is divided by 7? (a) 6 (b) 4 (c) 2 (d) 5
30. 30.Which of these numbers is divisible by 8? (a) 5738 (b) 5735 (c) 5751 (d) 5736
31. 31.Which of these numbers is divisible by 4? (a) 2100 (b) 2105 (c) 2099 (d) 2101
32. 32.What is the units digit of 4^53? (a) 4 (b) 5 (c) 6 (d) 3
33. 33.What is the units digit of 8^53? (a) 4 (b) 8 (c) 2 (d) 6
34. 34.What is the units digit of 2^27? (a) 4 (b) 6 (c) 8 (d) 2
35. 35.Which of these numbers is divisible by 3? (a) 2625 (b) 2626 (c) 2630 (d) 2624
36. 36.What is the remainder when 127 is divided by 5? (a) 4 (b) 1 (c) 3 (d) 2
37. 37.What is the remainder when 106 is divided by 3? (a) 1 (b) 0 (c) 2
38. 38.What is the smallest number you must add to 381 to make it divisible by 11? (a) 7 (b) 3 (c) 4 (d) 5
39. 39.What is the smallest number you must add to 506 to make it divisible by 10? (a) 3 (b) 5 (c) 6 (d) 4
40. 40.What is the remainder when 709 is divided by 11? (a) 7 (b) 6 (c) 4 (d) 5

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