Divisor Count of 2025

Example of Use of Divisor Count Function

$\map {\sigma_0} {2025} = 15$

where $\sigma_0$ denotes the divisor count function.

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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:

$2025 = 3^4 \times 5^2$

Thus:

\(\ds \map {\sigma_0} {2025}\)

\(=\)

\(\ds \map {\sigma_0} {3^4 \times 5^2}\)

\(\ds \)

\(=\)

\(\ds \paren {4 + 1} \paren {2 + 1}\)

\(\ds \)

\(=\)

\(\ds 15\)

The divisors of $2025$ can be enumerated as:

$1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 135, 225, 405, 675, 2025$

$\blacksquare$