ykaiim wrote:

If K is a positive integer, how many different prime numbers are factors of the expression \(K^2\)?

1) Three different prime numbers are factors of \(4K^4\).

2) Three different prime numbers are factors of 4K.

We need to determine the number of different prime factors in K^2. We must remember that the number of distinct prime factors of K^n for any positive integer n is same as the number of distinct prime factors of K. Thus, K^2 has the same distinct prime factors as K. That is, if we know the number of distinct prime factors K has, then we know the number of distinct prime factors K^2 has.

Statement One Alone:

Three different prime numbers are factors of 4K^4.

The information in statement one is not sufficient.

For instance, if K = 2 x 3 x 5, then 4K^4 has 3 different prime factors and K^2 also has 3 different prime factors (since K has 3 different prime factors). However, if K = 3 x 5, then 4K^2 has 3 different prime factors; however, K^2 has 2 different prime factors (since K has 2 different prime factors).

Statement Two Alone:

Three different prime numbers are factors of 4K.

The information in statement two is not sufficient.

For instance, if K = 2 x 3 x 5, then 4K has 3 different prime factors and K^2 has 3 different prime factors. However, if K = 3 x 5, then 4K has 3 different prime factors; however, K has 2 different prime factors.

Statements One and Two Together:

We see that using both statements still allows for K to have 2 or 3 different prime factors.

Answer: E

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