Divisor Count of 720

Example of Use of Divisor Counting Function

 * $\map \tau {720} = 24$

where $\tau$ denotes the divisor counting (tau) function.

Proof
From Divisor Counting Function from Prime Decomposition:
 * $\ds \map \tau n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$

where:
 * $r$ denotes the number of distinct prime factors in the prime decomposition of $n$
 * $k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.

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

Thus:

The divisors of $720$ can be enumerated as:
 * $1$, $2$, $3$, $4$, $5$, $6$, $8$, $9$, $10$, $12$, $15$, $16$, $18$, $20$, $24$, $30$, $36$, $40$, $45$, $48$, $60$, $72$, $80$, $90$, $120$, $144$, $180$, $240$, $360$, $720$