a^2 - b^2 multiple of 4 or not ?

[blue.vikasbhardwaj2]

If a, b, and c are integers and abc ≠ 0, is a2 - b2 a multiple of 4?

(1) a = (c - 1)2

(2) b = c2 - 1

The Answer to the Question is C .

My Solution :

We need to check whether a^2 - b^ 2 = 4k , k =1,2,3,4,....

After solving for a^2 - b^2 we have

a^2 - b^2 = -4c * (c-1)^2

If we have a look , if c = 1 then a^2 - b^2 = 0 (a^2 - b^2 not a multiple of 4)

But if c = -2 then a^2 - b^2 = 72 ( a^2 - b^2 multiple of 4)

Therefore , the Answer should be E

Please let me know where am i going wrong !!!!

Thanks!
Vikas

[Rijul Negi]

Is a^2 - b^2 = 4k ?

(a-b)(a+b) => for this to be multiple of 4, a and b should both be either odd or even...


Statement 1;
Insufficient because it tells us nothing about b

Statement 2:
Insufficient because it tells us nothing about a


Combing the two statements:
Lets say c is even,
from statement 1, a is odd
from statement 2, b is odd
hence, (a -b)(a+b) is multiple of 4

Now lets say c is odd,
Statement 1 => a is even
Statement 2 => b is also even
(a-b)(a+b) is multiple of 4


Option C is the correct option.

[jnelson0612]

Thank you; this is a very nice explanation!