Wilson's Theorem/Examples/5 divides (5-1)!+1

Example of Use of Wilson's Theorem

$5$ is a divisor of $\paren {5 - 1}! + 1$.


Proof

For the first few $n$ we see:

\(\ds \paren {5 - 1}! + 1\) \(=\) \(\ds 4! + 1\)
\(\ds \) \(=\) \(\ds 24 + 1\)
\(\ds \) \(=\) \(\ds 25\)
\(\ds \) \(=\) \(\ds 5 \times 5\)

Hence $5 \divides \paren {5 - 1}! + 1$.

$\blacksquare$


Sources

  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Wilson's theorem
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Wilson's theorem
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