According to Fermat's little theorem, for any p is a prime integer and gcdx,p=1, then the congruence xp-1≡1mod n is true, if we remove the restriction that gcdx,p=1, we may declarexp-1≡xmod p. For every integer x. Euler extended Fermat's Theorem as follows: if gcdx,p=1,then,where xÏ•n≡1mod n.Ï• is Euler's phi-function.

Euler's theorem cannot be implemented for any every integers x in the same manners as Fermat’s theorem works; that is, the congruence   xÏ•n+1≡xmod n is not always true. In this paper, we discussed the validation of congruence xÏ•n+1≡xmod n.