TITLE: Can $18$ consecutive integers be separated into two groups,such that their product is equal?
QUESTION [68 upvotes]: Can $18$ consecutive positive integers be separated into two groups, such that their product is equal?  We cannot leave out any number and neither we can take any number more than once. 
My work:
When the smallest number is not  $17$ or its multiple, there cannot exist any such arrangement as $17$ is a prime.  
When the smallest number is a multiple of $17$ but not of $13$ or $11$, then no such arrangement exists.  
But what happens, when the smallest number is a multiple of $ 17 $ and $13$ or $11$ or both?
Please help!

REPLY [50 votes]: If $18$ consecutive positive integers could be separated into two groups with equal products, then the product of all $18$ integers would be a perfect square. However, the product of two or more consecutive positive integers can never be a perfect square, according to a famous theorem of P. Erdős, Note on products of consecutive integers, J. London Math. Soc. 14 (1939), 194-198.