Highly Composite Number/Examples/60

Example of Highly Composite Number
$60$ is a highly composite number, being the smallest positive integer with $12$ divisors or more.

Proof
From Tau Function of 60:
 * $\map \tau {60} = 12$

For a positive integer $n$ to have a $\tau$ of $12$, the product of $1$ greater than the multiplicities of all the prime factors must be $12$.

The divisors of $12$ are $1, 2, 3, 4, 6, 12$.

The smallest $n$ with one prime factor such that $\map \tau n = 12$ is $2^{11} = 2048$.

The smallest $n$ with two prime factors such that $\map \tau n = 12$ are:
 * $2^3 3^2 = 72$
 * $2^5 3^1 = 96$

The smallest $n$ with three prime factors such that $\map \tau n = 12$ is:
 * $2^2 3^1 5^1 = 60$