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# How many different prime numbers are factors of the positive integer n ?

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 numbers are factors of n2.

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These ones can get a bit nasty.
(1) From the statement, it says four different prime numbers are factors of 2n. But it is not clear whether 2 is one of these prime numbers. Because if n is even, the presence of 2n would not add a prime factor. But in case of odd n, 2 would act as a supplement prime factor. So the number of prime factors change with even or odd n.
For Example.
If n = 2 * 3 * 5 * 7 (here n has 4 prime factors), then 2n = 2 * 2 * 3 * 5 * 7. Here 2n has 4 prime factors.
If n = 3 * 5 * 7 (here n has 3 prime factors), then 2n = 2 * 3 * 5 * 7. Here 2n has 4 prime factors.
From the above examples, it can be seen that number of different prime numbers may vary; NOT sufficient.
(2) Four different prime numbers are factors of n² implies four different prime numbers are factors of n as well; SUFFICIENT.

answered Apr 13, 2015 by Guru (5,808 points)
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Jyotsna Mehta