A two-digit number $ab$ is multiplied by its reverse $ba$. The ones (units) and tens digits of the four-digit resultant number are both $0.$ What is the value of the smallest such two-digit number $ab ?$

$25$

Step by Step Explanation:
1. Since the units digit of the resultant number is zero, either $a$ or $b$ must be $5.$
Without loss of generality, assume $a = 5.$
Therefore, $b$ is even.
2. Since the answer ends in $00$
$\implies$ The answer is a multiple of $100$ and hence is a multiple of $25.$
Since $b \ne 0$ and $ba$ ends in $5, ba$ is a multiple of $25.$
The only $2$-digit multiples of $25$ ending in $5$ are $25$ and $75.$
From step 1, $b$ is even and $7$ is not an even number.
Therefore $ba = 25$
Therefore, the two possible values for $ab$ are $25$ and $52$.
3. Hence, the smallest value of the two-digit number $ab$ is $25.$