Prime Factors of 52 Factorial

Example of Factorial
The prime decomposition of $52!$ is given as:


 * $52! = 2^{49} \times 3^{23} \times 5^{12} \times 7^8 \times 11^4 \times 13^4 \times 17^3 \times 19^2 \times 23^2 \times 29 \times 31 \times 37 \times 41 \times 43 \times 47$

Proof
For each prime factor $p$ of $52!$, let $a_p$ be the integer such that:
 * $p^{a_p} \mathrel \backslash 52!$
 * $p^{a_p + 1} \nmid 52!$

Taking the prime factors in turn:

Similarly:


 * $a_{29} = 1$
 * $a_{31} = 1$
 * $a_{37} = 1$
 * $a_{41} = 1$
 * $a_{43} = 1$
 * $a_{47} = 1$

Hence the result.