We are looking for positive integers less than 30 which have no common prime factors with 30.
Prime factors of 30 = 2, 3, 5
Numbers between 1 and 29 with no common prime factor = 1, 7, 11, 13, 17, 19 , 23, 29
That a total of 8 positive integers.

Answer is option D

swerve wrote:How many positive integers less than 30 have no common prime factor with 30?

A. 5
B. 6
C. 7
D. 8
E. 9

The OA is D.

Please, can anyone explain this PS question? I tried to solve it but I can't get the correct answer. I need help. Thanks.

Hi swerve.

The first step is to find the prime factors of 30. This can be done as follows: $$30=2\cdot15=2\cdot3\cdot5.$$ So, the prime factors of 30 are 2, 3 and 5.

Now, from 1 to 29, we must discard all even numbers, since 2 is a factor of all of them.

Also, we must discard all the multiples of 3 and all the multiples of 5.

Finally, we only have the numbers: 1, 7, 11, 13, 17, 19, 23 and 29.

So, there are 8 numbers that hold the condition.

The correct answer is the option D.

I hope it helps.

Regards.

swerve wrote:How many positive integers less than 30 have no common prime factor with 30?

A. 5
B. 6
C. 7
D. 8
E. 9

We need to determine the number of positive integers less than 30 that are relatively prime to 30. That is, the greatest common factor of each of those numbers and 30 is 1. We can see that these numbers are:

1, 7, 11, 13, 17, 19, 23, 29

So there are 8 such numbers.

Answer: D