A Question about SOAP

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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