I have a problem with this question
John writes the number 1!, 2!, 3!, ... , 199!, 200! on a whiteboard. John then erases one of the numbers. John then multiplied the remaining 199 numbers. He found out that the number was a perfect square. What was the number that was erased?
Can someone explain?
To be a perfect square, every prime factor has to appear an even number of times. $199$ is prime and is a factor of $199!$ and $200!$, so it appears an even number of times in the original product. If you erased $199!$ or $200!$ the product could not be square as it would have an odd number of $199$s. $197$ is prime. Which factorials does it appear in? It also restricts which factorials can be erased. What prime is the largest one that appears an odd number of times in the product? Are there others? You might be interested in De Polignac's formula for the exact power of a prime dividing a factorial. It is also known as Legendre's formula if you are searching.