The integers from 1 to 1000 are written in order around a circle. Starting at 1,

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The integers from 1 to 1000 are written in order around a circle. Starting at 1, every fifteenth number is marked (that is 1, 16, 31, etc.). This process is continued until a number is reached which has already been marked. How many unmarked numbers remain?

Answer: Option D
:
D
Consider the first round: - All the numbers which leave a remainder of 1 when divided by 15 will be marked. (1, 16, 31, 46 ... 991)
Consider the second round: - The first number to be marked is 991 + 15 - 1000 = 6. Thereafter all the numbers which leave a remainder 6 when divided by 15 will be marked. (6, 21, 36, 51 ... 996)
Consider the third round: - The first number to be marked is 996 + 15 - 1000 = 11. And all the numbers which leave a remainder 11 when divided by 15 will be marked. (11, 26, 41, 56 ... 986)
The first number to be marked in the fourth round is 986 + 15 - 1000 = 1. Now the cycle will repeat.
If we see the numbers which are getting marked are: 1, 6, 11, 16, 21, 26, 31, 36 ... 996
i.e. all the numbers which divided by 5 leave a remainder 1. So there are

\frac{1000}{5} = 200

numbers. So, 1000 - 200 = 800 numbers remain unmarked.

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