If n and t are positive integers, what is the greatest prime factor of nt?

[BTGmoderatorDC]

If n and t are positive integers, what is the greatest prime factor of nt?

(1) The greatest common factor of n and t is 5
(2) The least common multiple of n and t is 105


OA B

Source: GMAT Prep

[swerve]

If n and t are positive integers, what is the greatest prime factor of nt?

(1) The greatest common factor of n and t is 5
(2) The least common multiple of n and t is 105


OA B

Source: GMAT Prep

From statement 1, we have
- It's possible that \(n = t = 5.\) Then the greatest prime factor of \(nt\) is \(5.\)
- It's possible that \(n = 5\) and \(t = 35.\) Then the greatest prime factor of \(nt\) is \(7.\)
Not Sufficient \(\Large{\color{red}\chi}\)

From statement 2
The least common multiple contains every factor of \(t\) or \(n\) at least once (it has to; if, say, \(t\) had a factor that wasn't contained in it, then it would fail to be a multiple of \(t.\)) so, the biggest prime factor of this \(\#\) will also be the biggest prime factor of the product \(nt.\)
Sufficient \(\Large{\color{green}\checkmark}\)

Therefore, B