1. ## Alternative methods

I'm looking for alternative methods a question. My answer relies on a bit of guesswork and I don't think it's elegant, so I'm seeking other answers.
I'll post my solution in another post

Question
Let x and y be integers. Find the value of $3x^2y^2$ if
$y^2+3x^2y^2=30x^2+517$

2. My solution

$y^2+3x^2y^2=30x^2+517$

$y^2-517=3x^2(10-y^2)$

Fact 1) The L.H.S. must be a multiple of 3
Fact 2) If 0<y\leq{3}, then the R.H.S. is positive, but the L.H.S. is -ive, hence y>3
Fact 3) If y>22, a similar situation occurs, so 3<y<22
Fact 4) y cannot be a multiple of 3, as the L.H.S. would not be multiple of 3
Fact 5) I have stopped determining constraints and begun to guess the answer

Upon guessing, y=7 and x=2
So the value of $3x^2y^2$ follows

3. I think,

if you had 30x^2 + 517 could you not take x^2 to the other side?

and if you did and then divided (y^2) + (3x^2)(y^2)/(x^2)

Would that not cancel out the x2?

Giving you y^2 + (3)(y^2) = 30 + 517
??

I am probably breaking all kinds of rules but it seems that you loose x in that case and then you would have

4y^2 = 537

y^2 = 537/4

y = square root of (537/4)

I know I am all kinds of wrong I was just curious... Its ok to make a bit of fun of me I dont mind :P

Brian

4. Hello, I-Think!

Let $x$ and $y$ be integers.
Find the value of $3x^2y^2$ if: . $y^2+3x^2y^2\:=\:30x^2+517$

We have: . $y^2(1+3x^2) \:=\:30x^2+517 \quad\Rightarrow\quad y^2 \:=\:\frac{30x^2+517}{3x^2+1} \quad\Rightarrow\quad y^2\;=\;10 + \frac{507}{3x^2+1}$ .[1]

Since $y^2$ is an integer, $3x^2+1$ must be a factor of 507.

The factors of 507 are: . $1,\:3,\:13,\:39,\:169,\:507$

The only case in which $x^2$ is an integer is: . $3x^2+1 \:=\:13 \quad\Rightarrow\quad x^2 \,=\,4$

Sustitute into [1]: . $y^2 \:=\:10 + \frac{507}{3(4)+1} \quad\Rightarrow\quad y^2 \:=\:49$

Therefore: . $3x^2y^2 \:=\:3(4)(49) \:=\:588$