Special Highly Composite Number/Examples/60

Example of Special Highly Composite Number
$60$ 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, $60$ is highly composite.

$n > 60$ is a highly composite number which is not divisible by $60$.

We have that $12$ is a special highly composite number.

Therefore $12$ is a divisor of $n$.

As $60$ is not a divisor of $n$, it follows that while $3$ is a divisor of $n$, $5$ is not.

From Prime Decomposition of Highly Composite Number, no prime number greater than $5$ is a divisor of $n$.

Thus:
 * $n = 2^a \times 3^b$

where $a \ge b \ge 1$.

This will be investigated on a case-by-case basis.


 * $(1): \quad b = 1$

That is, $n = 2^a \times 3$.

We have that $n > 60$.

Therefore:
 * $(1 \text a): \quad a \ge 5$

as $2^4 \times 3^1 = 48$.

Then:

It follows by Proof by Contradiction that $b \ne 1$.


 * $(2): \quad b = 2$

That is, $n = 2^a \times 3^2$.

We have that $n > 60$.

Therefore:
 * $(2 \text a): \quad a \ge 3$

as $2^2 \times 3^2 = 36$.

Then:

It follows by Proof by Contradiction that $b \ne 2$.


 * $(3): \quad b \ge 3$

By Prime Decomposition of Highly Composite Number we have that $a \ge 3$.

Then:

By Proof by Cases it is seen that the existence of a highly composite $n$ not divisible by $60$ leads to a contradiction.

The result then follows by Proof by Contradiction.