# Divisor Count Function/Examples/12

< Divisor Count Function/Examples(Redirected from Divisor Count of 12)

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## Example of Use of Divisor Count Function

- $\map {\sigma_0} {12} = 6$

where $\sigma_0$ denotes the divisor count function.

## Proof

From Divisor Count Function from Prime Decomposition:

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

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:

- $12 = 2^2 \times 3$

Thus:

\(\ds \map {\sigma_0} {12}\) | \(=\) | \(\ds \map {\sigma_0} {2^2 \times 3^1}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \paren {2 + 1} \paren {1 + 1}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 6\) |

The divisors of $12$ can be enumerated as:

- $1, 2, 3, 4, 6, 12$

$\blacksquare$