${\displaystyle \alpha$ !} denotes the factorial of ${\displaystyle \alpha }$ . For all natural numbers, it gives the product of all numbers less than or equal to ${\displaystyle \alpha }$ .

Examples

5 is a prime number because,

${\displaystyle 4!=24}$

and

${\displaystyle 24\equiv -1{\pmod {5}}\implies 25|5}$

which is true. 6, on the other hand, is not, as

${\displaystyle 5!=120}$

and

${\displaystyle 120\equiv -1{\pmod {6}}\implies 121|6}$

which is false.