Problem:
Let a and b be relatively prime integers with a>b>0 and
(a−b)3a3−b3=373
What is a−b?
Answer Choices:
A. 1
B. 2
C. 3
D. 4
E. 5
Solution:
Note that
(a−b)3a3−b3=a2−2ab+b2a2+ab+b2
Hence the given equation may be written as 3a2+3ab+3b2=73a2−146ab+73b2. Combining like terms and factoring gives (10a−7b)(7a−10b)=0. Because a>b, and a and b are relatively prime, a=10 and b=7. Thus a−b=(C)3.
The problems on this page are the property of the MAA's American Mathematics Competitions