M23-35

Official Solution:

If \(x\) is a positive integer, \(f(x)\) is defined as the number of positive integers which are less than \(x\) and do not have a common factor with \(x\) other than 1. If \(x\) is prime, then \(f(x)=\)?

A. \(x - 2\)
B. \(x - 1\)
C. \(\frac{(x + 1)}{2}\)
D. \(\frac{(x - 1)}{2}\)
E. 2


Basically the question is: how many positive integers are less than given prime number \(x\), which has no common factor with \(x\) except 1.

Well, as \(x\) is a prime, all positive numbers less than \(x\) have no common factors with \(x\) (except common factor 1). So there would be \(x-1\) such numbers (as we are looking number of integers less than \(x\)).

For example consider \(x=7=prime\): how many numbers are less than 7 and have no common factors with 7: 1, 2, 3, 4, 5, and 6, so total of \(7-1=6\) numbers.


Answer: B

Hi Bunuel,

Great explanation and a great question. Thanks.

Just for the sake of understanding, could you please let me know what would be the answer to the question if we were given x = 36 instead of x = prime number?

Please find my solution below.

To calculate the number of positive integers less than 36 which do not have a common factor with 36 other than 1 we can do manual counting, in which case I got 11 numbers (5 7 11 13 17 19 23 25 29 31 35).

Could you validate this and let me know if there is any other (a sophisticated) approach?

Thanks again!

Hi Bunuel, can you explain to me what would have happened if you used 9 in your example as opposed to 7? Thanks.