Problem:
The fraction
(32007)2−(32005)2(32008)2−(32006)2
simplifies to which of the following?
Answer Choices:
A. 1
B. 49
C. 3
D. 29
E. 9
Solution:
First note that
(32007)2−(32005)2(32008)2−(32006)2=92007−9200592008−92006
Factoring 92005 from each of the terms on the right side produces
92007−9200592008−92006=92005⋅92−92005⋅192005⋅93−92005⋅91=9200592005⋅92−193−9=9⋅92−192−1=(E)9
The problems on this page are the property of the MAA's American Mathematics Competitions