Product of digits of a 2-digit number is 72. If we add 9 to the number, the new number obtained is a number formed by interchange of the digits. Find the number.

Let the tens digit of the number be x and the units digit be y.
The number can be written as 10x+y.

Given that the product of the digits is 72:

x×y=72— (Equation 1)
When 9 is added to the number, the digits are reversed (10y+x):

(10x+y)+9=10y+x
10x−x+y−10y+9=0
9x−9y=−9
Dividing by 9:

x−y=−1⟹y=x+1— (Equation 2)
Substitute Equation 2 into Equation 1:

x(x+1)=72
x
2
+x−72=0
(x+9)(x−8)=0
Since x is a digit, it must be a positive single digit integer, so x=8.

Now find y:

y=8+1=9
Thus, the number is 10(8)+9=89.