Data Sufficiency

srcc25anu wrote:See image attached for detailed calculations.

Ans C

Brent@GMATPrepNow wrote:

acchi369 wrote:If a, b, and c are integers and abc ≠ 0, is a^2 - b^2 a multiple of 4?

(1) a = (c - 1)^2

(2) b = c^2 - 1

NOTE: I added some exponent signs (^) to the question to avoid any confusion.

Target question: Is a^2 - b^2 a multiple of 4?

Since a^2 - b^2 = (a + b)(a - b), we can rephrase the target question . . .

Rephrased target question: Is (a + b)(a - b) a multiple of 4?

Statement 1: a = (c - 1)^2
This tells us a little bit about the properties of a, but we know nothing about b. So, there is no way to determine whether or not (a + b)(a - b) is a multiple of 4.
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: b = c^2 - 1
This tells us a little bit about the properties of b, but we know nothing about a. So, there is no way to determine whether or not (a + b)(a - b) is a multiple of 4.
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Statements 1 and 2 combined:
Statement 1 tells us that a = (c - 1)^2
Expand to get: a = c^2 - 2c + 1

Statement 2 tells us that b = c^2 - 1

No we can replace a and b with their respective values:
(a + b)(a - b) = [(c^2 - 2c + 1) + (c^2 - 1)] [(c^2 - 2c + 1) - (c^2 - 1)]
= [2c^2 - 2c][-2c + 2]
= [2(c^2 - c)][2(-c + 1)] I factored the 2 out of each set of brackets
= 4(c^2 - c)(-c + 1)
At this point, we can see that (a + b)(a - b) is definitely a multiple of 4.
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C


Cheers,
Brent