Oh, I love a good math puzzle! Let's solve this, shall we?
Let's assume the tens digit is 'a' and the ones digit is 'b'. We know that the product of the digits is 24, so we have the equation a * b = 24.
Now, we also know that when the digits are reversed, the new number is greater than the original number by 18. This means that the new number is equal to the original number plus 18, or (10*b + a) = (10*a + b) + 18.
Simplifying that equation gives us 9*b - 9*a = 18, which further simplifies to b - a = 2.
Now, we have two equations: a * b = 24 and b - a = 2. Let's solve them simultaneously.
Since b - a = 2, we can substitute b = a + 2 into the first equation, a * (a + 2) = 24. Expanding this equation gives us a^2 + 2a - 24 = 0.
Using the quadratic formula, we find that a = 4 or a = -6. Since we're looking for a positive two-digit number, we can discard the negative value.
So, a = 4. Substituting that back into the equation b - a = 2 gives us b - 4 = 2, which means b = 6.
Therefore, the number is 46.