Prime Factors of 39 Factorial

Example of Factorial
The prime decomposition of $39!$ is given as:


 * $39! = 2^{35} \times 3^{18} \times 5^8 \times 7^5 \times 11^3 \times 13^3 \times 17^2 \times 19^2 \times 23 \times 29 \times 31 \times 37$

Proof
For each prime factor $p$ of $39!$, let $a_p$ be the integer such that:
 * $p^{a_p} \mathrel \backslash 39!$
 * $p^{a_p + 1} \nmid 39!$

Taking the prime factors in turn:

Similarly:


 * $a_{23} = 1$
 * $a_{29} = 1$
 * $a_{31} = 1$
 * $a_{37} = 1$

Hence the result.