hari003 wrote:
How many different prime numbers are factors of positive integer n?
(1) 4 different prime numbers are factors of 2n
(2) 4 different prime numbers are factors of n^2
Dear
hari003,
I'm happy to help.
We want to know the number of different prime number factors of n. In other words, when we take the
prime factorization of n, how many primes do we see?
Statement #1:
4 different prime numbers are factors of 2nThis is a tricky one. If n = (3)(5)(7) = 105, then 2n = 210 would have four different prime number factors, {2, 3, 5, 7}. With this example, n started with just three different prime number factors.
By contrast, if n = (2)(3)(5)(7) = 210, then 2n = 420 would have four different prime number factors, {2, 3, 5, 7}. With this example, n started with four different prime number factors.
With this statement, it's possible for n to start with either three different prime number factors (not including 2) or with four different prime number factors (including 2). Because we can give two different answers to the prompt question, this statement, alone and by itself, is
not sufficient.
Statement #2:
4 different prime numbers are factors of n^2When we square a number, we change the total number of factors, but no new prime number factors are created in the process of squaring. It is impossible for (n^2) to have a prime factor that is not also a factor of n. Thus, however many different prime number factors n has, (n^2) has to have the exact same number. If (n^2) has four different prime number factors, then it absolutely must be true that n has four different prime number factors. This statement allows us to give a definitive numerical answer to the prompt question. This statement, alone and by itself, is
sufficient.
First statement is not sufficient, second statement is. OA =
(B).
Does all this make sense?
Mike
_________________
Mike McGarry
Magoosh Test PrepEducation is not the filling of a pail, but the lighting of a fire. — William Butler Yeats (1865 – 1939)