Problem: If both x and y are integers, how many pairs of solutions are there of the equation (x−8)(x−10)=2y?
Answer Choices:
A. 0
B. 1
C. 2
D. 3
E. more than 3
Solution:
(x−8)(x−10)=2y,x2−18x+80−2y=0
∴x=218±(18)2−4(80−2y)=9±1+2y
For x to be an integer, 1+2y must be the square of an integer. This is true only for y=3, and for y=3,x=12 or 6 .
There are, therefore, two solutions, namely, (12,3) and (6,3).