ardz24 wrote:If p is positive, is p prime?
(1) p^3 has exactly 4 distinct factors
(2) p^2 - p - 6 = 0.
What's the best way to determine which statement is sufficient?
(1) p^3 has exactly 4 distinct factors.
Case 1: Assume that p is prime, thus, the number of factors of p^3 are 1, p, p^2, and p^3. There are four factors.
Case 2: Assume that p is non-prime; let's assume = 4, a convenient value. The number of factors of 4^3 (= 64) are 1, 2, 4, 8, 16, 32, and 64. The number of factors is not four, thus, this case is invalid.
In case, you wish to try Case 1 with some values such as 2 and 101, you may. For p = 2, the number 2^3 has 1, 2, 4, and 8 as factors (four); for p = 10, the number 101^3 has 1, 101, 101^2, and 101^3 as factors (four).
Sufficient.
(2) p^2 - p - 6 = 0
=> p^2 - 3p + 2p - 6 = 0 => p(p - 3) + 2(p - 3) = 0 => p = 3 or -2. Since p is positive, p = 3, a prime number. Sufficient.
The correct answer:
D
Hope this helps!
-Jay
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