In the definition of Carmichael number, why is it necessary to have $(b, n) = 1$?

Question

In number theory, a Carmichael number is a composite number $n$ which satisfies the modular arithmetic congruence relation $$b^{n-1}\equiv 1\pmod{n}$$

for all integers $1<b<n$ which are relatively prime to $n$.

In the definition of Carmichael number, why is it necessary to have $(b,n) = 1$?

I need to understand this point, please.

Answer 1

Any prime that divides both $b$ and $n$ will also divide $b^{n-1}$, making it impossible to have $b^{n-1} \equiv 1 \mod n$.