Special Highly Composite Number/Examples/12

Example of Special Highly Composite Number
$12$ is a special highly composite number, being a highly composite number which is a divisor of all larger highly composite numbers.

Proof
By inspection of the sequence of highly composite numbers, $12$ is highly composite.

$n > 12$ is a highly composite number which is not divisible by $12$.

We have that $6$ is a special highly composite number.

Therefore $6$ is a divisor of $n$.

As $12$ is not a divisor of $n$, it follows that the multiplicity of $2$ in $n$ is $1$.

From Prime Decomposition of Highly Composite Number, that means:
 * $n = 2 \times 3 \times 5 \times r$

where $r$ is a possibly vacuous square-free product of prime numbers strictly greater than $5$.

Then:

The result follows by Proof by Contradiction.