If N is a positive integer...

Hello.

If you see carefully the options on the list you can determine which is the correct answer.

Let's see option by option.

- Option A: 30
30=2*3*5, so 30 has 3 different prime factors. OUT

- Option B: 60
60=2*2*3*5, so 60 has 3 different prime factors. OUT.

- Option C: 180
180=2*2*3*3*5, so 180 has 3 different prime factors, OUT.

- Option D: 210
210=2*3*5*7, so 210 has exactly 4 different prime factors. In addition, 14*210/60=49, so 14N/60 is an integer. CORRECT.

We don't need to analyze the last option.

So, the answer is D.

I hope this answer could help you.

I'm available if youd like any follow up.

LUANDATO wrote:If N is a positive integer and 14N/60 is an integer. What is the smallest value of N for which N has exactly four different prime factors?

A)30
B)60
C)180
D)210
E)cannot be determined

14N/60 = 7N/30.

The smallest prime numbers are 2, 3, 5 and 7.
Thus, the smallest integer with 4 different prime factors = 2*3*5*7 = 210.
Test whether N=210 yields an integer value for 7N/30:
(7*210)/30= 7*7 = 49.
Success!
Thus, the least possible value for N = 210.

The correct answer is D.

Hello,

Excellent, your explanations was helpful to me a lot! Thank you so much!