Question stem, rephrased:M7MBA wrote:Is p^2 - 1 divisible by 12?
(1) p > 3
(2) p is a prime number
Is (p+1)(p-1) divisible by 12?
Statement 1:
Case 1: p=4
In this case, (p+1)(p-1) = 5*3 = 15, so the answer to the question stem is NO.
Case 2: p=5
In this case, (p+1)(p-1) = 6*4 = 24, so the answer to the question stem is YES.
Since the answer is NO in Case 1 but YES in Case 2, INSUFFICIENT.
Statement 2:
Case 2 also satisfies Statement 2.
In Case 2, the answer to the question stem is YES.
Case 3: p=2
In this case, (p+1)(p-1) = 3*1 = 3, so the answer to the question stem is NO.
Since the answer is NO in Case 3 but YES in Case 2, INSUFFICIENT.
Statements combined:
If p=5, then (p+1)(p-1) = 6*4.
If p=7, then (p+1)(p-1) = 8*6.
If p=11, then (p+1)(p-1) = 12*10.
If p=13, then (p+1)(p-1) = 14*12.
If p=17, then (p+1)(p-1) = 18*16.
Notice the PATTERN:
In every case, the product in blue is composed of a MULTIPLE OF 3 and a MULTIPLE OF 4 or includes a factor that is a MULTIPLE OF BOTH 3 AND 4.
Since (p+1)(p-1) is divisible by both 3 and 4, it must be divisible by 12.
Thus, the answer to the question stem is YES.
SUFFICIENT.
The correct answer is C.
