Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 529 · 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.Which of these numbers is divisible by 3? (a) 1300 (b) 1298 (c) 1303 (d) 1299
2. 2.What is the remainder when 176 is divided by 3? (a) 0 (b) 1 (c) 2
3. 3.What is the smallest number you must add to 152 to make it divisible by 7? (a) 3 (b) 5 (c) 1 (d) 2
4. 4.What is the smallest number you must add to 182 to make it divisible by 6? (a) 4 (b) 3 (c) 5 (d) 2
5. 5.What is the smallest number you must add to 160 to make it divisible by 3? (a) 2 (b) 1
6. 6.Which of these numbers is divisible by 4? (a) 1479 (b) 1471 (c) 1473 (d) 1472
7. 7.What is the smallest number you must add to 532 to make it divisible by 10? (a) 2 (b) 7 (c) 8 (d) 9
8. 8.What is the smallest number you must add to 317 to make it divisible by 10? (a) 7 (b) 4 (c) 2 (d) 3
9. 9.Which of these numbers is divisible by 11? (a) 4653 (b) 4655 (c) 4648 (d) 4674
10. 10.What is the units digit of 7^45? (a) 7 (b) 3 (c) 1 (d) 9
11. 11.Which of these numbers is divisible by 9? (a) 642 (b) 656 (c) 639 (d) 634
12. 12.What is the units digit of 3^17? (a) 7 (b) 1 (c) 9 (d) 3
13. 13.What is the smallest number you must add to 793 to make it divisible by 10? (a) 3 (b) 7 (c) 6 (d) 8
14. 14.What is the units digit of 4^17? (a) 3 (b) 5 (c) 4 (d) 6
15. 15.What is the smallest number you must add to 31 to make it divisible by 3? (a) 2 (b) 1
16. 16.Which of these numbers is divisible by 11? (a) 7711 (b) 7725 (c) 7704 (d) 7718
17. 17.What is the units digit of 4^52? (a) 6 (b) 7 (c) 4 (d) 5
18. 18.Which of these numbers is divisible by 9? (a) 480 (b) 486 (c) 487 (d) 497
19. 19.Which of these numbers is divisible by 8? (a) 2681 (b) 2675 (c) 2680 (d) 2695
20. 20.What is the remainder when 441 is divided by 8? (a) 2 (b) 0 (c) 1 (d) 7
21. 21.What is the remainder when 794 is divided by 11? (a) 3 (b) 2 (c) 1 (d) 9
22. 22.Which of these numbers is divisible by 6? (a) 2658 (b) 2669 (c) 2654 (d) 2661
23. 23.What is the units digit of 7^36? (a) 1 (b) 9 (c) 3 (d) 7
24. 24.What is the units digit of 7^33? (a) 3 (b) 9 (c) 7 (d) 1
25. 25.What is the smallest number you must add to 450 to make it divisible by 7? (a) 2 (b) 6 (c) 4 (d) 5
26. 26.What is the smallest number you must add to 148 to make it divisible by 6? (a) 4 (b) 3 (c) 1 (d) 2
27. 27.What is the units digit of 4^36? (a) 5 (b) 7 (c) 6 (d) 4
28. 28.What is the units digit of 8^58? (a) 6 (b) 8 (c) 2 (d) 4
29. 29.What is the smallest number you must add to 431 to make it divisible by 12? (a) 2 (b) 1 (c) 11 (d) 3
30. 30.Which of these numbers is divisible by 9? (a) 4073 (b) 4062 (c) 4051 (d) 4059
31. 31.What is the remainder when 236 is divided by 5? (a) 4 (b) 2 (c) 0 (d) 1
32. 32.What is the units digit of 3^50? (a) 1 (b) 9 (c) 3 (d) 7
33. 33.Which of these numbers is divisible by 11? (a) 3913 (b) 3905 (c) 3898 (d) 3923
34. 34.What is the smallest number you must add to 201 to make it divisible by 5? (a) 1 (b) 4 (c) 3 (d) 2
35. 35.Which of these numbers is divisible by 9? (a) 7202 (b) 7189 (c) 7191 (d) 7199
36. 36.What is the smallest number you must add to 628 to make it divisible by 9? (a) 3 (b) 1 (c) 2 (d) 7
37. 37.What is the remainder when 132 is divided by 8? (a) 3 (b) 5 (c) 4 (d) 6
38. 38.Which of these numbers is divisible by 8? (a) 4202 (b) 4195 (c) 4200 (d) 4214
39. 39.What is the remainder when 890 is divided by 11? (a) 8 (b) 9 (c) 1 (d) 10
40. 40.Which of these numbers is divisible by 8? (a) 2184 (b) 2188 (c) 2193 (d) 2180

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