Divisor Count of 625

Example of Use of Divisor Count Function

$\map {\sigma_0} {625} = 5$

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:

$625 = 5^4$

Thus:

\(\ds \map {\sigma_0} {625}\)

\(=\)

\(\ds \map {\sigma_0} {5^4}\)

\(\ds \)

\(=\)

\(\ds 4 + 1\)

\(\ds \)

\(=\)

\(\ds 5\)

The divisors of $625$ can be enumerated as:

$1, 5, 25, 125, 625$

$\blacksquare$