ThmDex – An index of mathematical definitions, results, and conjectures.

Result R5115 on D6: Natural number factorial function

Natural number factorial is an even number for numbers 2 or greater

Formulation 0

Let $N \in \{ 2, 3, 4, \ldots \}$ be a D5094: Positive integer.

Then $N !$ is an D5: Even integer.

Proofs

Proof 0

Let $N \in \{ 2, 3, 4, \ldots \}$ be a D5094: Positive integer.

We can write \begin{equation} \begin{split} N ! & = N (N - 1) (N - 2) \cdots 4 \cdot 3 \cdot 2 \cdot 1 \\ & = 2 \left( N (N - 1) (N - 2) \cdots 4 \cdot 3 \cdot 1 \right) \\ & = 2 k \end{split} \end{equation} for $k = N (N - 1) (N - 2) \cdots 4 \cdot 3 \cdot 1$. $\square$