For a positive integer $n$ greater than $1$, define the ** s**um

For example, $\text{Soap}(2) = (1 \times 2) = 2$ $\text{Soap}(3) = (1 \times 2) + (1 \times 3) + (2 \times 3) = 11$

If the arithmetic mean of the products is an integer, then $n$ is defined as "Soapy".

For example,

- $2$ is "Soapy" since $\text{Soap}(2) = 2$ and there is $1$ pair and $\frac21$ is an integer
- $3$ is not "Soapy" since $\text{Soap}(3) = 11$ and there are $3$ pairs and $\frac{11}{3}$ is not an integer

Are numbers of the form $10^k$ "Soapy", where $k \in \mathbb{N}$?