Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 762 · 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 226 is divided by 11? (a) 8 (b) 7 (c) 6 (d) 5
2. 2.What is the remainder when 252 is divided by 5? (a) 4 (b) 2 (c) 1 (d) 3
3. 3.What is the remainder when 165 is divided by 10? (a) 7 (b) 5 (c) 6 (d) 4
4. 4.What is the remainder when 493 is divided by 7? (a) 4 (b) 3 (c) 2 (d) 5
5. 5.Which of these numbers is divisible by 6? (a) 3125 (b) 3130 (c) 3118 (d) 3120
6. 6.Which of these numbers is divisible by 8? (a) 6673 (b) 6669 (c) 6685 (d) 6672
7. 7.What is the remainder when 308 is divided by 5? (a) 2 (b) 3 (c) 1 (d) 4
8. 8.What is the units digit of 7^39? (a) 1 (b) 3 (c) 7 (d) 9
9. 9.Which of these numbers is divisible by 4? (a) 169 (b) 166 (c) 164 (d) 163
10. 10.What is the units digit of 3^43? (a) 1 (b) 7 (c) 3 (d) 9
11. 11.Which of these numbers is divisible by 4? (a) 593 (b) 592 (c) 591 (d) 598
12. 12.Which of these numbers is divisible by 11? (a) 8154 (b) 8140 (c) 8138 (d) 8143
13. 13.Which of these numbers is divisible by 3? (a) 2035 (b) 2039 (c) 2032 (d) 2034
14. 14.What is the smallest number you must add to 423 to make it divisible by 6? (a) 2 (b) 5 (c) 4 (d) 3
15. 15.What is the smallest number you must add to 502 to make it divisible by 8? (a) 6 (b) 2 (c) 1 (d) 3
16. 16.Which of these numbers is divisible by 11? (a) 9863 (b) 9880 (c) 9867 (d) 9874
17. 17.What is the units digit of 3^27? (a) 3 (b) 1 (c) 7 (d) 9
18. 18.What is the smallest number you must add to 247 to make it divisible by 5? (a) 2 (b) 4 (c) 3 (d) 1
19. 19.What is the smallest number you must add to 367 to make it divisible by 12? (a) 4 (b) 5 (c) 7 (d) 6
20. 20.What is the units digit of 2^51? (a) 6 (b) 8 (c) 2 (d) 4
21. 21.What is the smallest number you must add to 85 to make it divisible by 4? (a) 2 (b) 3 (c) 1
22. 22.What is the units digit of 4^50? (a) 5 (b) 4 (c) 7 (d) 6
23. 23.What is the units digit of 7^21? (a) 3 (b) 7 (c) 1 (d) 9
24. 24.Which of these numbers is divisible by 11? (a) 5431 (b) 5434 (c) 5435 (d) 5454
25. 25.What is the remainder when 575 is divided by 9? (a) 6 (b) 7 (c) 8 (d) 1
26. 26.What is the units digit of 8^51? (a) 4 (b) 6 (c) 8 (d) 2
27. 27.What is the smallest number you must add to 371 to make it divisible by 10? (a) 1 (b) 9 (c) 7 (d) 8
28. 28.Which of these numbers is divisible by 9? (a) 3230 (b) 3232 (c) 3241 (d) 3231
29. 29.What is the remainder when 620 is divided by 12? (a) 9 (b) 4 (c) 8 (d) 7
30. 30.Which of these numbers is divisible by 8? (a) 3420 (b) 3424 (c) 3435 (d) 3427
31. 31.What is the remainder when 398 is divided by 12? (a) 3 (b) 1 (c) 2 (d) 10
32. 32.What is the smallest number you must add to 233 to make it divisible by 5? (a) 3 (b) 4 (c) 2 (d) 1
33. 33.What is the units digit of 3^10? (a) 7 (b) 9 (c) 3 (d) 1
34. 34.Which of these numbers is divisible by 8? (a) 2724 (b) 2728 (c) 2729 (d) 2741
35. 35.What is the units digit of 3^20? (a) 9 (b) 3 (c) 7 (d) 1
36. 36.What is the remainder when 402 is divided by 5? (a) 2 (b) 3 (c) 4 (d) 1
37. 37.What is the smallest number you must add to 434 to make it divisible by 12? (a) 11 (b) 10 (c) 9 (d) 2
38. 38.Which of these numbers is divisible by 8? (a) 6027 (b) 6038 (c) 6024 (d) 6018
39. 39.What is the units digit of 3^23? (a) 3 (b) 9 (c) 1 (d) 7
40. 40.Which of these numbers is divisible by 3? (a) 344 (b) 349 (c) 345 (d) 346

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