How many different prime numbers are factors of the positive integer n? 1) four different prime numbers are factors of 2n 2) four different prime umbers are factors of n^2.


Stmt 1 :
it's possible that 2 is already a factor of n to start with, in which case n itself would still have 4 different prime factors (because, in that case, the additional 2 would not change the total number of prime factors).
for instance, if n = 3x5x7 = 105 (which has three prime factors), then 2n = 2x3x5x7 = 210 has four prime factors. if n = 2x3x5x7 = 210, which has four prime factors, then 2n = 2x2x3x5x7 = 420, which still has two prime factors. therefore, #1 is not sufficient.
Stmt 2 :
We know that for n^2 will contain all the prime numbers on n raise to the power 2. Hence it is sufficient.