Prime Factors of 13 Factorial

Example of Factorial
The prime decomposition of $13!$ is given as:


 * $13! = 2^{10} \times 3^5 \times 5^2 \times 7 \times 11 \times 13$

Proof
For each prime factor $p$ of $13!$, let $a_p$ be the integer such that:
 * $p^{a_p} \mathrel \backslash 13!$
 * $p^{a_p + 1} \nmid 13!$

Taking the prime factors in turn:

Similarly:


 * $a_{11} = 1$
 * $a_{13} = 1$

Hence the result.