# Math Help - Find all three digit even numbers N such that 693 times N is a perfect square.

1. ## Find all three digit even numbers N such that 693 times N is a perfect square.

Find all three digit even numbers N such that 693 times N is a perfect square.

2. Originally Posted by vonge
Find all three digit even numbers N such that 693 times N is a perfect square.

First of all observe that $693 = 9 .11 . 7$. So if $693N$ is a perfect square, then $77|N$ since only one factor of 7 and 11. More clearly $N = 77k^2$.

Now choose even k such that $77k^2$ is a three digit number

You will see that for k= 4, N is already a four digit number. Thus k=2 is the only answer.

Or $N = 77 (2)^2 = 308$

So the only three digit that satisfies is $N = 308$

3. Thank you so much for that