Factor factorials

Question

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!

Answer 1

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 $79$

Answer 2

Hint: Since $82! = 1\cdot 2\cdot 3\cdots 81\cdot 82$, every prime number that divides $81$ must divide one of the numbers $1,2,3,4\dots 82$.

Answer 3

For $n!$ look for the largest prime less than or equal to $n$. For example, for $82!$, I'd likely start at 82 and work down until I found a prime which would be 79.

Consider a small $n$ like 5 for a moment. $5!=120$ where I'd claim that 5 is the greatest prime divisor of 120 as $120=5*4*3*2*1$ and thus the greatest prime will just be the largest prime in the sequence which is 5.