<s>[INST] Which two digit number when you find the product of the digits yields a number that is half the original? Which two digit number when you find the product of the digits yields a number that is half the original?
Let x=$ab$ be the $2$-digit number. So $x=10a+b$.
Then $ab=\frac{x}{2} \implies ab=\frac{10a+b}{2} \implies 2ab=10a+b \implies b=\frac{10a}{2a-1}$. I guess $a=3$ and get $b=6$. So the answer is $36$.
But how can this be done without guessing.
 [/INST] From what you figured out so far, $ab=5a+\frac{b}{2}$. If there is an integer solution, $b$ is even and greater than $5$, so $b$ is either $6$ or $8$. Also (see the picture), $\frac{b}{2}$ is divisible by $b-5$, which rules out $b=8$. If $b=6$, then the rightmost rectangle below is $1$ unit across and has area $\frac{6}{2}=3$, so $a=3$.

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