Problem Solving

If the sum of two positive integers is 24 and the difference of their squares is 48, what is the product of the two integers

(A) 108
(B) 119
(C) 128
(D) 135
(E) 143

We can let x = one integer and y = the other integer and create the following equations:

x + y = 24

and

x^2 - y^2 = 48

Notice that x^2 - y^2 = (x + y)(x - y), and so:

(x + y)(x - y) = 48

Since x + y = 24:

24(x - y) = 48

x - y = 2

So, we have x + y = 24 and x - y = 2. If we add these two equations together, we have 2x = 26 or x = 13. Once we know x = 13, y must be 11 since x + y = 24. Thus, xy = (13)(11) = 143.

Answer: E