Multiplicity of 720 in 720 Factorial

Theorem
The multiplicity of $720$ in $720!$ is $178$.

That is:
 * $720^{178} \mathrel \backslash 720!$

but:
 * $720^{179} \nmid 720!$

where:
 * $720!$ denotes $720$ factorial
 * $\backslash$ denotes divisibility
 * $\nmid$ denotes non-divisibility.

Proof
We have that:
 * $720 = 2^4 \times 3^2 \times 5$

It remains to inspect the divisibility of $2$, $3$ and $5$ in $720!$

Thus:

Multiplicity of $5$ in $720!$
We calculate the multiplicity of the powers of $2$ and $3$ in $720!$ thus:

Thus it is seen that the smallest power of the prime powers that are divisors of $720$ that divide $720!$ is that of $3^2$, which is $168$.

Hence:
 * $720^{168} \mathrel \backslash 720!$

but:
 * $720^{169} \nmid 720!$