Divisor Count of 42

Example of Use of $\tau$ Function

 * $\tau \left({42}\right) = 8$

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:
 * $42 = 2 \times 3 \times 7$

Thus:
 * $\tau \left({42}\right) = \tau \left({2^1 \times 3^1 \times 7^1}\right) = \left({1 + 1}\right) \left({1 + 1}\right) \left({1 + 1}\right) = 8$

The divisors of $12$ can be enumerated as:
 * $1, 2, 3, 6, 7, 14, 21, 42$