opponent 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.
(1) If K = 30, then K has 3 different prime factors (2, 3, 5), 4K^4 has 3 different prime factors (2, 3, 5), K² also has 3 different prime factors (2, 3, 5).
If K = 21, then K has 2 different prime factors (3, 7), but 4K^4 has 3 different prime factors (2, 3, 7), K² has 2 different prime factors (3, 7).
No unique answer.
So, (1) is NOT SUFFICIENT.
(2) If K = 30, then K has 3 different prime factors (2, 3, 5), 4K has 3 different prime factors (2, 3, 5), K² also has 3 different prime factors (2, 3, 5).
If K = 21, then K has 2 different prime factors (3, 7), but 4K has 3 different prime factors (2, 3, 7), K² has 2 different prime factors (3, 7).
No unique answer.
So, (2) is NOT SUFFICIENT.
Combining (1) and (2) also we don't get a unique answer. So, again NOT SUFFICIENT.
The correct answer is
E.
