vinay1983 wrote:If "x" and "y" are positive integers such that x > y, what is the remainder when x^4 - y^4 is divided by 4?
1. When x-y is divided by 4, the remainder is 0
2. When x+y is divided by 4, the remainder is 1
Target question: What is the remainder when x^4 - y^4 is divided by 4?
IMPORTANT: Notice that we can factor x^4 - y^4.
x^4 - y^4 = (x² + y²)(x² - y²)
= (x² + y²)(x + y)(x - y)
So, we can now rephrase the target question . . .
REPHRASED target question: What is the remainder when (x² + y²)(x + y)(x - y) is divided by 4?
Statement 1: When x-y is divided by 4, the remainder is 0
In other words,
(x-y) is a multiple of 4.
This means that (x² + y²)
(x + y)(x - y) must also be a multiple of 4.
In other words,
when (x² + y²)(x + y)(x - y) is divided by 4, the remainder must be 0
Since we can answer the
target question with certainty, statement 1 is SUFFICIENT
Statement 2: When x+y is divided by 4, the remainder is 1
HMMMMMMMMMMMMMMMMMMMMM, this statement contradicts the statement 1. That's a problem, since we must assume that each statement is true.
In statement 1, we know that x - y is a multiple of 4. This means that x-y is EVEN.
If x-y is even, then either x and y are both even, or x and y are both odd.
In statement 2, we're told that we get a remainder of 1 when x+y is divided by 4. This means that x+y is ODD.
If x+y is even, then EITHER x is odd and y is even, OR x is even and y is odd.
So, as you can see, statement 2 contradicts statement 1.
At this point, I'm going to stop my solution.
Cheers,
Brent
