Multiplicity of 720 in 720 Factorial

Theorem

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

That is:

$720^{178} \divides 720!$

but:

$720^{179} \nmid 720!$

where:

$720!$ denotes $720$ factorial

$\divides$ 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 $2$ in $720!$

The prime factor $2$ appears in $720!$ to the power of $716$.

That is:

$2^{716} \divides 720!$

but:

$2^{717} \nmid 720!$

Multiplicity of $3$ in $720!$

The prime factor $3$ appears in $720!$ to the power of $356$.

That is:

$3^{356} \divides 720!$

but:

$3^{357} \nmid 720!$

Multiplicity of $5$ in $720!$

The prime factor $5$ appears in $720!$ to the power of $178$.

That is:

$5^{178} \divides 720!$

but:

$5^{179} \nmid 720!$

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

\(\ds 716\)

\(=\)

\(\ds 4 \times 179\)

\(\ds \leadsto \ \ \)

\(\ds \paren {2^4}^{179}\)

\(\divides\)

\(\ds 720!\)

\(\ds 356\)

\(=\)

\(\ds 2 \times 178\)

\(\ds \leadsto \ \ \)

\(\ds \paren {3^2}^{178}\)

\(\divides\)

\(\ds 720!\)

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 $178$.

Hence:

$720^{178} \divides 720!$

but:

$720^{179} \nmid 720!$

$\blacksquare$

Sources

• 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $720$