The product of the digits of a positive two-digit number exceeds the sum of the digits by 39. If the order of the digits is reversed, the number is increased by 27. Find the number.
To solve this problem, let's first assume that the tens digit is represented by the variable 'x' and the units digit by the variable 'y'.
According to the problem, the product of the digits exceeds the sum by 39. This can be expressed as:
xy = x + y + 39.
We also know that when the digits are reversed, the number is increased by 27. This can be expressed as:
10y + x = 10x + y + 27.
Now we have a system of two equations:
xy = x + y + 39,
10y + x = 10x + y + 27.
To solve this system of equations, we can use substitution or elimination method.
Let's use the substitution method:
From the first equation, we have xy = x + y + 39.
Rearranging this equation, we get:
xy - x - y = 39.
Factoring out the common variables, we get:
x(y - 1) - (y - 1) = 39.
Simplifying further, we get:
(x - 1)(y - 1) = 39.
Now, let's list down the factors of 39: 1, 3, 13, and 39.
We can substitute these values into the equation (x - 1)(y - 1) = 39 and check which combinations give valid solutions.
Case 1:
If x - 1 = 1 and y - 1 = 39, we get x = 2 and y = 40, which is not a valid solution since the units digit cannot be 40.
Case 2:
If x - 1 = 3 and y - 1 = 13, we get x = 4 and y = 14.
Case 3:
If x - 1 = 13 and y - 1 = 3, we get x = 14 and y = 4.
Case 4:
If x - 1 = 39 and y - 1 = 1, we get x = 40 and y = 2, which is not a valid solution.
Therefore, the valid solutions are x = 4 and y = 14 or x = 14 and y = 4.
So, the two-digit number can be either 41 or 14.
Hence, the number is 41 or 14.