Factor factorials How would you find the greatest prime factor of a factorial? For instance, 82! The 2 and 41 that are yielded when you prime-factor 82 seem to have no correlation to the prime factorization of 82!

You know that $$82!=82\times 81\times 80\times \dots \times 3\times 2\times 1$$ What is the greatest prime factor? Well first, we can find the biggest prime number in the expansion of $82!$. That prime is $79$. Is there another prime factor greater than that? I can tell you that any factors greater than $79$ are composite numbers. $80=2\times 40$, $81=3\times 27$, and $82=2\times 41$. Any factors greater than $82$ are a product of two or more factors in the expansion of $82!$; they must be composite numbers.

Therefore the greatest prime factor in $82!$ is $79$