Divisible by 4

[cesar.rodriguez.blanco]

I do not know how to deal with this DS. Please, help!!!:

If p, x, and y are positive integers, y is odd, and p = x^2 + y^2, is x divisible by 4?

(1) When p is divided by 8, the remainder is 5.
(2) x - y = 3

[Ben Ku]

Please cite the source (author) of this problem. We cannot reply unless a source is cited (and, if no source is cited, we will have to delete the post!). Thanks.

[cesar.rodriguez.blanco]

The source is
http://blog.clearadmit.com/category/gmat-tips/

[Ben Ku]

I think this problem is as challenging as any of the MGMAT Challenge Archive problems! I will tackle only (1) because that is the toughest part of this problem to evaluate.

Here's another approach to the problem different from the solution given. The question asks, "Is x divisible by 4?" which we can rephrase as "Is x^2 divisible by 16?"

We know that p = x^2 + y^2, so x^2 = p - y^2. Since y is odd, y^2 is also odd. If p is even, then the statement is sufficient to answer the question.

Statement (1) says that when p is divided by 8, it has a remainder of 5. We can rewrite p = 8n + 5 (where n is an integer). Here, p is clearly odd, so we are still uncertain of whether the statement is sufficient.

Because y is odd, we can represent y = 2k + 1 (where k is an integer), because it's an odd number. Let's plug these expressions for p and y into our original equation:

x^2 = p - y^2
x^2 = (8n + 5) - (2k + 1)^2
x^2 = 8n + 5 - (4k^2 + 4k + 1)
x^2 = 8n + 5 - 4k^2 - 4k - 1
x^2 = 8n - 4k^2 - 4k - 4
x^2 = 4(2n - k^2 - k - 1)

Up to this point, we know that x^2 is a multiple of 4. In order for x^2 to be a multiple of 16, (2n - k^2 - k - 1) must be a multiple of 4.

Now I'll group the k's together.
x^2 = 4[(2n - 1) - (k^2 + k)]
x^2 = 4[(8n - 1) - k(k + 1)]

Here, clearly (8n - 1) is odd, and k(k + 1) is even. An odd number minus an even number gives an odd number.

x^2 = 4[ODD - EVEN] = 4[ODD].

We have found out that x^2 is not divisible by 16, so x is not divisible by 4. This statement is sufficient to answer the question, and the answer is (A).

The key to my approach (which can be translated to other similar problems) is to restate "when p is divided by 8, the remainder is 5" as "p = 8n + 5" and to rephrase "y is odd" as "y = 2k + 1".

Hope that helps.