Problem Solving

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

The OA is D.

Why D is the correct answer? Can any expert explain this PS question please?. Thanks.

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!