A two-digit number is such that the product of the digits is $8$. When $18$ is added to the number, then the digits are reversed. The number is:

Let the two-digit number be represented as $10x+y$, where $x$ and $y$ are the digits.
Given that the product of the digits is $8$:

$$x\cdot y=8$$

When $18$ is added to the number, the digits are reversed:

$$(10x+y)+18=10y+x$$

$$18=9y-9x$$

$$9(y-x)=18⟹y-x=2⟹y=x+2$$

Substitute $y=x+2$ into the product equation:

$$x(x+2)=8$$

$$x^2+2x-8=0$$

$$(x+4)(x-2)=0$$

Since $x$ must be a positive single digit, we have $x=2$.
Then, $y=2+2=4$.

The number is $10(2)+4=24$.