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.

I notice that stopping to analyze the question first before going straight to the statements make it easier to solve this DS questions.

The question is looking for the number of distinct prime numbers of K. Whether it be K^4, K^100 or K^9, the number of distinct prime numbers would be the same with just K.

Statement 1: 3 prime number factors for 4K^4. Well there are two possibilites:

(a) either K has "2" and 2 other prime numbers (e.g. 2, 3 and 7; 2, 5 and 11) OR

(b) K just have two prime numbers and no "2" (e.g. 5 and 7)

Thus, we know K could have 2 or 3 distinch prime number factors. INSUFFICIENT.

Statement 2: Once you get used to it, you will notice Statement (2) is just the same as Statement (1).. It throws in "2" outside the K making it blurry whether "2" is within K or not. Thus, INSUFFICIENT.

If Statement (1) and Statement (2) are just similar givens. Then together, they are INSUFFICIENT.

Answer: E

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Impossible is nothing to God.