Divisor Count Function/Examples/105

Example of Use of $\tau$ Function

 * $\tau \left({105}\right) = 8$

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:
 * $105 = 3 \times 5 \times 7$

Thus:
 * $\tau \left({105}\right) = \tau \left({3 \times 5 \times 7}\right) = \left({1 + 1}\right) \left({1 + 1}\right) \left({1 + 1}\right) = 8$

The divisors of $105$ can be enumerated as:
 * $1, 3, 5, 7, 15, 21, 35, 105$