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

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• May 24th 2008, 09:07 PM
vonge
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.

I have no idea how to go about this?

Please help urgently!
• May 24th 2008, 10:11 PM
Isomorphism
Quote:

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

I have no idea how to go about this?

Please help urgently!

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 $N = 308$
• May 28th 2008, 01:25 AM
vonge
Thank you so much for that