Products of Consecutive Integers in 2 Ways

Theorem

The following integers are the product of consecutive integers in $2$ ways:

$-720, 720, 5040$

Proof

From 720 is Product of Consecutive Numbers in Two Ways:

$720 = 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8$

From 5040 is Product of Consecutive Numbers in Two Ways:

$5040 = 7 \times 6 \times 5 \times 4 \times 3 \times 2 = 10 \times 9 \times 8 \times 7$

Then:

$-720 = \left({-6}\right) \left({-5}\right) \left({-4}\right) \left({-3}\right) \left({-2}\right) = \left({-10}\right) \left({-9}\right) \left({-8}\right)$

The same trick cannot be used for $-5040$ because there are four divisors in $10 \times 9 \times 8 \times 7$, and negating them makes the product positive.

Hence the result.

$\blacksquare$