Factorial creates Prime

Number Theory Level 1

There is a number $N>2$ such that the digits of $N!$ can be rearranged to form a prime .

Now, which of the following option is correct?

$A :$ There exists infinitely many numbers $N$ .
$B :$ There exist only one such number $N$ .
$C :$ All prime numbers greater than $2$ satisfy the conditions of $N$ .
$D :$ There is no number satisfying the condition of $N$

C D B A

1 solution

Danish Ahmed
Jul 5, 2015

If $N > 3$ then $N!$ is divisible by $3$ , so the sum of its digits is divisible by $3$ , before and after rearrangement, so any number formed by rearranging its digits is still divisible by $3$ .

Thus, no qualifying $N$ exists.

Factorial creates Prime

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