Problem:
Which of the following numbers is a perfect square?
Answer Choices:
A. 98!⋅99!
B. 98!⋅100!
C. 99!⋅100!
D. 99!⋅101!
E. 100!⋅101!
Solution:
Note that for m<n we have
m!⋅n!=(m!)2⋅(m+1)⋅(m+2)⋯⋅⋅n
Therefore m!⋅n! is a perfect square if and only if
(m+1)⋅(m+2)⋯⋅⋅n
is a perfect square. For the ve answer choices, that quantity is
99,99⋅100,100,100⋅101, and 101,
and of those only 100 is a perfect square. Therefore the answer is 99!⋅100!.
Answer: C.
The problems on this page are the property of the MAA's American Mathematics Competitions