Is x^2+y^2 divisible by 5?

First I must say this is not GMAT question. As every GMAT divisibility question will tell you in advance that any unknowns represent positive integers.

But let's deal with question as it's written. x^2+y^2 to be divisible by 5 it should be an integer first and then should have the last digit 0 or 5:

(1) \(x-y=5n+1\) --> \((x-y)^2=25n^2+10n+1\) --> \(x^2+y^2=25n^2+10n+1+2xy\). Now 25n^2+10n is divisible by 5, but what about 1-2xy? Not sufficient.

(2) \(x+y=5m+3\) --> \((x+y)^2=25m^2+30m+9\) --> \(x^2+y^2=25m^2+30m+9-2xy\). The same here. Not sufficient.

(1)+(2) Add the equations:
\(2(x^2+y^2)=25n^2+10n+25m^2+30m+10=25(n^2+m^2)+10(n+3m)+10\).

\(x^2+y^2=\frac{25(n^2+m^2)}{2}+5(n+3m)+5\). Well if \(n^2+m^2\) is even, \(x^2+y^2\) is an integer and it's divisible by 5. But what if \(n^2+m^2\) is odd, then \(x^2+y^2\) is not an integer at all, hence not divisible by 5.

Can we somehow conclude that \(n^2+m^2\) is even? If we are not told that x and y are integers then not.

Answer: E.

To check this consider following pairs of x an y: x=4.5, y=3.5 and x=7, y=1,
\(x=4.5\), \(y=3.5\)
\(x-y=4.5-3.5=1\) first statement holds true, 1 divided by 5 remainder 1.
\(x+y=4.5+3.5=8\) second statement holds true, 8 divided by 5 remainder 3.
\(x^2+y^2=4.5^2+3.5^2=32.5\) not divisible by 5.

\(x=7\), \(y=1\)
\(x-y=7-1=6\) first statement holds true, 6 divided by 5 remainder 1.
\(x+y=7+1=8\) second statement holds true, 8 divided by 5 remainder 3.
\(x^2+y^2=7^2+1^2=50\) is divisible by 5.

Two different answers. Not sufficient. Answer: E.


Now let's see if we were told that x and y are positive integers, as GMAT does. Still would be quite hard 700+ question.

Last step:
\(x^2+y^2=\frac{25(n^2+m^2)}{2}+5(n+3m)+5\). Can we now conclude that that \(n^2+m^2\) is even?

\(x-y=5n+1\) and \(x+y=5m+3\). Add them up: \(2x=5(n+m)+4\) --> \(x=\frac{5(n+m)}{2}+2\). As x is an integer n+m must be divisible by 2, hence either both are even or both are odd, in any case \(n^2+m^2\) is even and divisible by 2. Hence \(x^2+y^2=\frac{25(n^2+m^2)}{2}+5(n+15m)+5\) is an integer and divisible by 5.

In this case answer C.

Hope it's clear.