Problem: If the sum is a perfect square and if is less than , then the possible values for are:
Answer Choices:
A. only 1
B. 1 and 8
C. only 8
D. 8 and 49
E. 1,8 , and 49
Solution:
. The possible values for are . For to be integral, the discriminant of the equation must be a perfect square. This fact reduces the possible values for to , and . Hence the values of are and .
Note: There are ways of shortening the number of trials for still further, but these involve a knowledge of number-theoretic theorems. The shortest way to do this problem is by testing the choices given.