Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 45 · 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 11? (a) 2803 (b) 2794 (c) 2789 (d) 2808
2. 2.What is the remainder when 191 is divided by 9? (a) 1 (b) 2 (c) 7 (d) 3
3. 3.Which of these numbers is divisible by 3? (a) 1246 (b) 1244 (c) 1241 (d) 1242
4. 4.What is the units digit of 4^13? (a) 4 (b) 5 (c) 3 (d) 6
5. 5.What is the smallest number you must add to 224 to make it divisible by 3? (a) 2 (b) 1
6. 6.What is the remainder when 230 is divided by 8? (a) 5 (b) 7 (c) 6 (d) 2
7. 7.What is the remainder when 399 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
8. 8.What is the remainder when 126 is divided by 12? (a) 6 (b) 5 (c) 8 (d) 7
9. 9.Which of these numbers is divisible by 6? (a) 2959 (b) 2966 (c) 2955 (d) 2958
10. 10.Which of these numbers is divisible by 6? (a) 1051 (b) 1065 (c) 1056 (d) 1061
11. 11.What is the smallest number you must add to 544 to make it divisible by 12? (a) 4 (b) 9 (c) 7 (d) 8
12. 12.What is the remainder when 477 is divided by 6? (a) 5 (b) 4 (c) 2 (d) 3
13. 13.What is the smallest number you must add to 310 to make it divisible by 7? (a) 4 (b) 2 (c) 5 (d) 6
14. 14.What is the units digit of 2^46? (a) 6 (b) 4 (c) 8 (d) 2
15. 15.What is the remainder when 360 is divided by 11? (a) 9 (b) 7 (c) 3 (d) 8
16. 16.What is the remainder when 109 is divided by 7? (a) 3 (b) 6 (c) 4 (d) 5
17. 17.What is the smallest number you must add to 434 to make it divisible by 6? (a) 4 (b) 2 (c) 5 (d) 3
18. 18.What is the units digit of 4^58? (a) 4 (b) 5 (c) 7 (d) 6
19. 19.Which of these numbers is divisible by 4? (a) 321 (b) 331 (c) 327 (d) 324
20. 20.Which of these numbers is divisible by 6? (a) 3311 (b) 3316 (c) 3304 (d) 3306
21. 21.Which of these numbers is divisible by 11? (a) 9320 (b) 9328 (c) 9340 (d) 9338
22. 22.Which of these numbers is divisible by 8? (a) 751 (b) 754 (c) 744 (d) 743
23. 23.What is the units digit of 2^16? (a) 2 (b) 6 (c) 4 (d) 8
24. 24.What is the remainder when 637 is divided by 9? (a) 6 (b) 2 (c) 7 (d) 8
25. 25.What is the remainder when 479 is divided by 8? (a) 1 (b) 6 (c) 5 (d) 7
26. 26.What is the remainder when 157 is divided by 3? (a) 1 (b) 0 (c) 2
27. 27.What is the smallest number you must add to 243 to make it divisible by 4? (a) 3 (b) 2 (c) 1
28. 28.Which of these numbers is divisible by 8? (a) 4531 (b) 4539 (c) 4550 (d) 4536
29. 29.Which of these numbers is divisible by 4? (a) 2634 (b) 2629 (c) 2626 (d) 2628
30. 30.What is the smallest number you must add to 505 to make it divisible by 7? (a) 6 (b) 5 (c) 4 (d) 1
31. 31.What is the units digit of 2^27? (a) 2 (b) 8 (c) 6 (d) 4
32. 32.What is the smallest number you must add to 70 to make it divisible by 3? (a) 1 (b) 2
33. 33.Which of these numbers is divisible by 4? (a) 1133 (b) 1130 (c) 1132 (d) 1137
34. 34.What is the smallest number you must add to 243 to make it divisible by 7? (a) 3 (b) 1 (c) 5 (d) 2
35. 35.What is the units digit of 9^18? (a) 2 (b) 9 (c) 0 (d) 1
36. 36.What is the remainder when 323 is divided by 10? (a) 3 (b) 7 (c) 2 (d) 4
37. 37.What is the smallest number you must add to 411 to make it divisible by 9? (a) 2 (b) 3 (c) 4 (d) 6
38. 38.What is the units digit of 4^51? (a) 3 (b) 6 (c) 4 (d) 5
39. 39.What is the smallest number you must add to 480 to make it divisible by 11? (a) 7 (b) 5 (c) 3 (d) 4
40. 40.What is the remainder when 172 is divided by 6? (a) 2 (b) 3 (c) 4 (d) 5

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