A number of two digits is equal to three times the product of its digits. The digit in the ten's place is 2 less than the digit in the unit's place. What is the number?

Let the ten's digit be $x$, so the unit's digit is $x+2$. Then the number is $10x+(x+2)$ and this equals $3x(x+2)$. So $11x+2=3x^2+6x$, giving $3x^2-5x-2=0=(x-2)(3x+1)$. Thus $x=2$ or $x=-\frac13$, but a digit cannot be negative or fractional. Hence the number is $24$.