# Divisor Count of 7560

## Example of Use of Divisor Count Function

$\map {\sigma_0} {7560} = 64$

where $\sigma_0$ denotes the divisor count function.

## Proof

$\ds \map {\sigma_0} n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$

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:

$7560 = 2^3 \times 3^3 \times 5 \times 7$

Thus:

 $\ds \map {\sigma_0} {7560}$ $=$ $\ds \map {\sigma_0} {2^3 \times 3^3 \times 5^1 \times 7^1}$ $\ds$ $=$ $\ds \paren {3 + 1} \paren {3 + 1} \paren {1 + 1} \paren {1 + 1}$ $\ds$ $=$ $\ds 64$

The divisors of $7560$ can be enumerated as:

$1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 27, 28, 30, 35, 36, 40, 42,$
$45, 54, 56, 60, 63, 70, 72, 84, 90, 105, 108, 120, 126, 135, 140, 168, 180, 189, 210,$
$216, 252, 270, 280, 315, 360, 378, 420, 504, 540, 630, 756, 840, 945, 1080, 1260, 1512, 1890, 2520, 3780, 7560$

$\blacksquare$