It has to do with the fact that does not exist unless a and n are coprime. Consider Def 1: . That means that . This algebraic step can only be done if you require that a and n are coprime.

So technically Def 1 does NOT imply Def 2 directly. The correct statement is actually: Def 1 (Def 2 OR )

Consider a concrete example. is a Carmichael number:

and as the theory suggests.

but because 3 and 561 are not coprime.