Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 151 · 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 remainder when 577 is divided by 9? (a) 8 (b) 2 (c) 0 (d) 1
2. 2.What is the remainder when 178 is divided by 6? (a) 5 (b) 2 (c) 4 (d) 3
3. 3.Which of these numbers is divisible by 3? (a) 732 (b) 730 (c) 736 (d) 733
4. 4.What is the smallest number you must add to 583 to make it divisible by 9? (a) 1 (b) 2 (c) 3 (d) 7
5. 5.What is the smallest number you must add to 43 to make it divisible by 4? (a) 1 (b) 3 (c) 2
6. 6.Which of these numbers is divisible by 3? (a) 2303 (b) 2301 (c) 2305 (d) 2300
7. 7.What is the remainder when 297 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 2
8. 8.What is the units digit of 4^27? (a) 3 (b) 5 (c) 6 (d) 4
9. 9.What is the units digit of 3^50? (a) 7 (b) 9 (c) 1 (d) 3
10. 10.What is the units digit of 2^46? (a) 6 (b) 2 (c) 4 (d) 8
11. 11.Which of these numbers is divisible by 4? (a) 1926 (b) 1921 (c) 1924 (d) 1929
12. 12.What is the remainder when 107 is divided by 6? (a) 3 (b) 5 (c) 4 (d) 1
13. 13.What is the smallest number you must add to 526 to make it divisible by 7? (a) 1 (b) 4 (c) 5 (d) 6
14. 14.Which of these numbers is divisible by 6? (a) 3521 (b) 3510 (c) 3508 (d) 3512
15. 15.Which of these numbers is divisible by 6? (a) 4936 (b) 4925 (c) 4930 (d) 4926
16. 16.What is the units digit of 9^44? (a) 2 (b) 1 (c) 9 (d) 0
17. 17.Which of these numbers is divisible by 3? (a) 2687 (b) 2681 (c) 2682 (d) 2684
18. 18.Which of these numbers is divisible by 3? (a) 2225 (b) 2223 (c) 2227 (d) 2222
19. 19.What is the smallest number you must add to 161 to make it divisible by 3? (a) 2 (b) 1
20. 20.What is the units digit of 8^54? (a) 8 (b) 4 (c) 2 (d) 6
21. 21.What is the smallest number you must add to 615 to make it divisible by 12? (a) 10 (b) 8 (c) 9 (d) 3
22. 22.What is the units digit of 2^19? (a) 6 (b) 4 (c) 8 (d) 2
23. 23.What is the units digit of 3^23? (a) 1 (b) 9 (c) 7 (d) 3
24. 24.Which of these numbers is divisible by 4? (a) 366 (b) 369 (c) 364 (d) 361
25. 25.Which of these numbers is divisible by 11? (a) 2913 (b) 2887 (c) 2893 (d) 2895
26. 26.Which of these numbers is divisible by 4? (a) 3431 (b) 3434 (c) 3427 (d) 3428
27. 27.What is the smallest number you must add to 258 to make it divisible by 11? (a) 7 (b) 6 (c) 8 (d) 5
28. 28.What is the remainder when 363 is divided by 6? (a) 2 (b) 3 (c) 5 (d) 4
29. 29.What is the smallest number you must add to 183 to make it divisible by 4? (a) 2 (b) 1 (c) 3
30. 30.Which of these numbers is divisible by 4? (a) 1194 (b) 1197 (c) 1191 (d) 1192
31. 31.What is the remainder when 205 is divided by 4? (a) 2 (b) 0 (c) 3 (d) 1
32. 32.What is the remainder when 604 is divided by 11? (a) 1 (b) 8 (c) 10 (d) 9
33. 33.Which of these numbers is divisible by 11? (a) 9034 (b) 9022 (c) 9020 (d) 9016
34. 34.What is the remainder when 464 is divided by 6? (a) 4 (b) 3 (c) 1 (d) 2
35. 35.What is the remainder when 203 is divided by 3? (a) 2 (b) 1 (c) 0
36. 36.What is the smallest number you must add to 496 to make it divisible by 12? (a) 9 (b) 7 (c) 8 (d) 4
37. 37.What is the units digit of 9^33? (a) 1 (b) 7 (c) 9 (d) 8
38. 38.What is the units digit of 8^35? (a) 8 (b) 2 (c) 4 (d) 6
39. 39.What is the units digit of 7^20? (a) 7 (b) 9 (c) 1 (d) 3
40. 40.What is the smallest number you must add to 555 to make it divisible by 11? (a) 7 (b) 8 (c) 5 (d) 6

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