A Question about SOAP

For a positive integer nn greater than 11, define the sum of all pairs: Soap(n)\text{Soap}(n) as the sum of all possible pair products, made of distinct integers from 11 to nn.

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

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

For example,

  • 22 is "Soapy" since Soap(2)=2\text{Soap}(2) = 2 and there is 11 pair and 21\frac21 is an integer
  • 33 is not "Soapy" since Soap(3)=11\text{Soap}(3) = 11 and there are 33 pairs and 113\frac{11}{3} is not an integer

Are numbers of the form 10k10^k "Soapy", where kNk \in \mathbb{N}?

×

Problem Loading...

Note Loading...

Set Loading...