Divisor Count of 224

Example of Use of $\tau$ Function

 * $\map \tau {224} = 12$

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

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

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:
 * $224 = 2^5 \times 7$

Thus:
 * $\map \tau {224} = \map \tau {2^5 \times 7^1} = \paren {5 + 1} \paren {1 + 1} = 12$

The divisors of $224$ can be enumerated as:
 * $1, 2, 4, 7, 8, 14, 16, 28, 32, 56, 112, 224$