Divisor Count of 64,000

From ProofWiki
Jump to navigation Jump to search

Example of Use of Divisor Count Function

$\map {\sigma_0} {64 \, 000} = 40$

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:

$64 \, 000 = 2^9 \times 5^3$


Thus:

\(\ds \map {\sigma_0} {64 \, 000}\) \(=\) \(\ds \map {\sigma_0} {2^9 \times 5^3}\)
\(\ds \) \(=\) \(\ds \paren {9 + 1} \paren {3 + 1}\)
\(\ds \) \(=\) \(\ds 40\)


The divisors of $64 \, 000$ can be enumerated as:

$1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100,$
$125, 128, 160, 200, 250, 256, 320, 400, 500, 512, 640, 800, 1000, 1280,$
$1600, 2000, 2560, 3200, 4000, 6400, 8000, 12800, 16000, 32000, 64000$

$\blacksquare$