# Math Help - Perfect squares

1. ## Perfect squares

Determine all positive integers n such that n+100 and n+168 are both perfect squares:
Here' my attempt:
n+168 should be less than 35sqr
Since, 35sqr - 34sqr = 69 which is greater than 68, the difference between n+100 and n+168

Beyond that, I've got no clue.

2. Originally Posted by darknight
Determine all positive integers n such that n+100 and n+168 are both perfect squares:
Here' my attempt:
n+168 should be less than 35sqr
Since, 35sqr - 34sqr = 69 which is greater than 68, the difference between n+100 and n+168

Beyond that, I've got no clue.
Hi darknight,

I would proceed by finding all qualifying perfect squares, then from them finding n.

$n+100=a^2$

$n+168=b^2$

So

$a^2-100=b^2-168$

$b^2-a^2=68$

$(b-a)(b+a)=68$

We know that $(b-a)$ and $(b+a)$ are integers, so this problem is now equivalent to factoring 68 into two divisors, $68=d_1d_2$.

For any pair $(d_1,d_2)$, we can solve

$(b-a) = d_1$

$(b+a) = d_2$

$\Rightarrow 2b = d_1+d_2$

First, try $68=1\cdot68$

$\Rightarrow 2b = 69$

No integer solutions.

Next try $68=2\cdot34$

$\Rightarrow 2b = 36$

$b = 18$

$a = 16$

$n = a^2 - 100 = 156$

Next try $68=4\cdot17$

Again there are no integer solutions.

So the only possible n is $\boxed{156}$.

3. Ah well, went till (b-a)(b+a) = 68 and then hit the brick wall..
Thanks