How many different prime numbers are factors of the positive

[sm]

Hi there,

DS problem for ya!

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.

the correct answer: B

Why isn't statement 1 sufficient? if we are told that four different prime numbers are factors of 2n then can't i further assume that one of those four prime numbers is 2 (since it's 2n) so wouldn't n have 3 different prime factors?? therefore sufficient info to answer the problem?

[RonPurewal]

if we are told that four different prime numbers are factors of 2n then can't i further assume that one of those four prime numbers is 2 (since it's 2n)

yes.

so wouldn't n have 3 different prime factors?? therefore sufficient info to answer the problem?

no.
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.