What is the greatest prime factor of $12! + 14!$?  (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\cdot 2\cdot 3\cdot \cdots \cdot (n-1)\cdot n$.)
To find the greatest prime factor of an expression, I need to factor it completely into primes and then take the largest one. One way to start factoring $12! + 14!$ is to notice that both terms have $12!$ as a factor, so I can factor it out. I get $12!(1 + 13\cdot 14)$. Now I need to factor $12!$ and $1 + 13\cdot 14$ further. $12!$ is just the product of all the numbers from $1$ to $12$, so I can write it as $2^{10}\cdot 3^5\cdot 5^2\cdot 7\cdot 11$. $1 + 13\cdot 14$ is $183$, which is not prime. I can divide $183$ by $3$ and get $61$, which is prime. So $183 = 3\cdot 61$. Therefore, the complete prime factorization of $12! + 14!$ is $2^{10}\cdot 3^6\cdot 5^2\cdot 7\cdot 11\cdot 61$. The largest prime factor in this factorization is $61$. 