Divisor Count of 140

Example of Use of $\tau$ Function

 * $\tau \left({140}\right) = 12$

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:
 * $140 = 2^2 \times 5 \times 7$

Thus:
 * $\tau \left({140}\right) = \tau \left({2^2 \times 5^1 \times 7^1}\right) = \left({2 + 1}\right) \left({1 + 1}\right) \left({1 + 1}\right) = 12$

The divisors of $140$ can be enumerated as:
 * $1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140$