Is K the square of an integer?

[billzhao]

If k is a positive integer, is k the square of an integer?
(1) k is divisible by 4
(2) k is divisible by exactly four different prime numbers

My answer is (B). Can anyone help to explain? Thanks.

[BuckeyeT]

What's the OA?
I'm getting (E). But, I'm not totally confident in my answer.

[billzhao]

My reasoning is that: If K is a product of four different prime numbers, then K will not be the square of any integers. Am I right?

[BuckeyeT]

That's actually what I was looking at in my answer. But as happens, I got caught up trying to prove it to be the square of an integer instead of proving if I can determine conclusively one way or the other.

By your logic, B sounds correct. I've even plugged a few values in, and it seems to hold.

[cjb]

If k is a positive integer, is k the square of an integer?
(1) k is divisible by 4
(2) k is divisible by exactly four different prime numbers

1) 20 is divisible by 4, but is not the square of an integer. 36 is divisible by 4, and is the square of an integer. INSUFFICIENT.

2) 210 (2*3*5*7) has four different prime factors, and is not the square of an integer. 210^2 also has four different prime factors (the same ones), and is the square of an integer. INSUFFICIENT.

1) + 2) together:

210^2 has four different prime factors, is a multiple of four, and is the square of an integer.

210 * 2 has four different prime factors, is a multiple of four, and is not the square of an integer. INSUFFICIENT

So it's E.

[Bidisha800]

(E)

[AJ2009]

I go with E