tarek99 wrote:
If k is a positive integer, is k the square of an integer?
(1) k is divisible by 4.
(2) k is divisble by exactly four different prime numbers.
Rephrase Question: Is k a perfect square?
(1)
If k = 4, the answer to the rephrase question is YES.
But if k = 8, the answer is NO.
So, INSUFFICIENT.
(2)
This means that k has four different prime factors, but we don't know how many times those factors appear in the prime factorization of k.
so, for example, if k is 2 x 3 x 5 x 7 (which is divisible by the four primes 2, 3, 5, 7), it's not the square of an integer(perfect square);
if k is 2 x 2 x 3 x 3 x 5 x 5 x 7 x 7 (divisible by the same four primes), it's the square of the integer 2 x 3 x 5 x 7.
so, INSUFFICIENT.
(together)
you need the 4 prime factors (because of statement 2), and you also need to have at least two '2's in the prime factorization (because of statement 1).
the aforementioned perfect square (2 x 2 x 3 x 3 x 5 x 5 x 7 x 7) still works.
to create a number that satisfies the criteria yet isn't a perfect square, just add another copy of one of these prime factors (e.g., 2 x 2 x 3 x 3 x 5 x 5 x 7 x 7 x 7).
INSUFFICIENT Together.
answer = e
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