For a positive integer $n$ greater than $1$, define the sum of all pairs: $\text{Soap}(n)$ as the sum of all possible pair products, made of distinct integers from $1$ to $n$.

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 $k \in \mathbb{N}$?

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