## Find T and M

$T \textup{ and }M$ in the following equation represent positive whole numbers where M represents the hundred’s digit of a six-digit number. Find the values of $T\textup{ and }M$.
$\left [3\left (230+T\right )\right ]^2=492M04$
Source: mathcontest.olemiss.edu 10/6/2008

SOLUTION
Since $492M04$ is a perfect square, we will check the 10 choices of $M=\left \{0,1,2,3,4,5,6,7,8,9\right \}$ to see which one will make it a perfect square as follows.
$\sqrt{492004}=701.43$
Not a perfect square.
$\sqrt{492104}=701.50$  Not
$\sqrt{492204}=701.57$  Not
$\sqrt{492304}=701.64$  Not
$\sqrt{492404}=701.72$  Not
$\sqrt{492504}=701.79$  Not
$\sqrt{492604}=701.86$  Not
$\sqrt{492704}=701.93$  Not
$\sqrt{492804}=702$  YES
$\sqrt{492904}=702.07$  Not
Thus, $M=8$.
Our equation now looks like this:
$\left [3\left (230+T\right )\right ]^2=492804$
$9\left (230+T\right )^2=492804$
$\left (230+T\right )^2=54756$
$230+T=\pm \sqrt{54756}$
$230+T=\pm 234$
$230+T=234$
because we are dealing with positive number
$T=234-230=4$.

Answer$T=4;\;M=8$.