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.

AnswerT=4;\;M=8.


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About mvtrinh

Retired high school math teacher.
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