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To prove k is an integer, for x < n:   \[S_{n-1}= \[\frac{S_{n} - 1}{S_{x}}=  \frac{S_{i}*S_{i-1}*S_{i-2}...S_{x}*S_{0}}{S_{x}}\] $S_{x}$ in the numerator and denominator will cancel out, and prove it is an integer.