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.