Number Theory Worksheet for Maths Olympiad

Maths Olympiad topic 19: Divisibility, remainders and units digits · Set 361 · 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 6? (a) 4990 (b) 5003 (c) 4993 (d) 4992
2. 2.What is the remainder when 375 is divided by 7? (a) 6 (b) 5 (c) 3 (d) 4
3. 3.What is the units digit of 8^27? (a) 2 (b) 4 (c) 6 (d) 8
4. 4.What is the remainder when 189 is divided by 6? (a) 4 (b) 3 (c) 5 (d) 2
5. 5.What is the remainder when 177 is divided by 8? (a) 1 (b) 0 (c) 7 (d) 2
6. 6.Which of these numbers is divisible by 4? (a) 2066 (b) 2075 (c) 2069 (d) 2068
7. 7.What is the units digit of 3^35? (a) 7 (b) 9 (c) 3 (d) 1
8. 8.Which of these numbers is divisible by 4? (a) 3394 (b) 3399 (c) 3403 (d) 3396
9. 9.What is the units digit of 8^23? (a) 8 (b) 6 (c) 2 (d) 4
10. 10.Which of these numbers is divisible by 8? (a) 2642 (b) 2657 (c) 2648 (d) 2655
11. 11.Which of these numbers is divisible by 4? (a) 2457 (b) 2456 (c) 2455 (d) 2463
12. 12.What is the remainder when 864 is divided by 11? (a) 6 (b) 7 (c) 8 (d) 5
13. 13.Which of these numbers is divisible by 8? (a) 1298 (b) 1296 (c) 1291 (d) 1307
14. 14.What is the remainder when 466 is divided by 8? (a) 1 (b) 6 (c) 3 (d) 2
15. 15.What is the smallest number you must add to 530 to make it divisible by 9? (a) 8 (b) 3 (c) 2 (d) 1
16. 16.What is the units digit of 9^18? (a) 0 (b) 1 (c) 2 (d) 9
17. 17.What is the smallest number you must add to 291 to make it divisible by 5? (a) 2 (b) 3 (c) 4 (d) 1
18. 18.What is the units digit of 2^29? (a) 4 (b) 8 (c) 2 (d) 6
19. 19.What is the remainder when 203 is divided by 10? (a) 4 (b) 7 (c) 3 (d) 2
20. 20.Which of these numbers is divisible by 11? (a) 7883 (b) 7871 (c) 7862 (d) 7865
21. 21.Which of these numbers is divisible by 3? (a) 626 (b) 622 (c) 624 (d) 629
22. 22.What is the smallest number you must add to 643 to make it divisible by 8? (a) 4 (b) 3 (c) 6 (d) 5
23. 23.What is the smallest number you must add to 370 to make it divisible by 6? (a) 3 (b) 4 (c) 1 (d) 2
24. 24.Which of these numbers is divisible by 8? (a) 6708 (b) 6696 (c) 6691 (d) 6703
25. 25.What is the units digit of 7^57? (a) 1 (b) 3 (c) 9 (d) 7
26. 26.What is the units digit of 9^42? (a) 1 (b) 0 (c) 2 (d) 9
27. 27.Which of these numbers is divisible by 8? (a) 6024 (b) 6036 (c) 6025 (d) 6017
28. 28.What is the remainder when 264 is divided by 7? (a) 2 (b) 4 (c) 5 (d) 6
29. 29.What is the smallest number you must add to 115 to make it divisible by 6? (a) 3 (b) 5 (c) 1 (d) 4
30. 30.Which of these numbers is divisible by 4? (a) 3203 (b) 3206 (c) 3198 (d) 3200
31. 31.What is the smallest number you must add to 321 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
32. 32.What is the smallest number you must add to 268 to make it divisible by 5? (a) 1 (b) 3 (c) 2 (d) 4
33. 33.What is the remainder when 963 is divided by 12? (a) 3 (b) 9 (c) 4 (d) 2
34. 34.Which of these numbers is divisible by 3? (a) 2048 (b) 2046 (c) 2045 (d) 2051
35. 35.What is the remainder when 389 is divided by 5? (a) 3 (b) 1 (c) 4 (d) 2
36. 36.What is the remainder when 563 is divided by 8? (a) 5 (b) 2 (c) 4 (d) 3
37. 37.What is the units digit of 9^34? (a) 2 (b) 0 (c) 9 (d) 1
38. 38.What is the remainder when 233 is divided by 3? (a) 2 (b) 0 (c) 1
39. 39.What is the remainder when 225 is divided by 4? (a) 1 (b) 3 (c) 0 (d) 2
40. 40.Which of these numbers is divisible by 3? (a) 138 (b) 139 (c) 136 (d) 143

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