Use Fermat's theorem to show that for any positive integer n, the integer

is divisible by 383838.

__My attempt:__
383838=(37)(19)(13)(7)(3)(2)

If 37 does not divide n, then by FLT,

. If, however, 37 divides n, then

. Thus in any case,

.

If 19 does not divide p, then by FLT,

. If, however, 19 divides n, then

. Thus in any case,

.

If 13 does not divide p, then by FLT,

. If, however, 37 divides n, then

. Thus in any case,

.

If 7 does not divide p, then by FLT,

. If, however, 7 divides n, then

. Thus in any case,

.

If 3 does not divide p, then by FLT,

. If, however, 3 divides n, then

. Thus in any case,

.

If 2 does not divide p, then by FLT,

. If, however, 2 divides n, then

. Thus in any case,

.

Combining these 6 congruences, we get

Could someone please check my proof? Thanks in advance.