A $2$-digit number is such that the product of its digits is $24$. If $18$ is subtracted from the number, the digits interchange their places. Find the number.

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Step 1: Let unit's digit be $y$ and ten's digit be $x$. The number is $10x + y$. $xy = 24$ ... (i), $(10x + y) - 18 = 10y + x \Rightarrow x - y = 2$ ... (ii)
Step 2: From (i) and (ii): $x^2 - 2x - 24 = 0$
Step 3: $\Rightarrow (x - 6)(x + 4) = 0$
Step 4: $x = 6$, $y = 4$. The number is $64$.

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