Divisor Count of 64,000

Example of Use of $\tau$ Function

 * $\tau \left({64 \, 000}\right) = 40$

where $\tau$ denotes the $\tau$ Function.

Proof
From Tau Function from Prime Decomposition:
 * $\displaystyle \tau \left({n}\right) = \prod_{j \mathop = 1}^r \left({k_j + 1}\right)$

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:

The divisors of $64 \, 000$ can be enumerated as:
 * $1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 64, 80, 100,$
 * $125, 128, 160, 200, 250, 256, 320, 400, 500, 512, 640, 800, 1000, 1280,$
 * $1600, 2000, 2560, 3200, 4000, 6400, 8000, 12800, 16000, 32000, 64000$