elementary number theory - Question regarding Fermat's little theorem

Fermat's little theorem says:

If $p$ is prime, and $a$ is an integer with $p \nmid a$, then $a^{p - 1 } \equiv 1 \pmod{p}$.

And this is a small part that I extracted from the book:

Let $n = 4k + r$ with $0 \leq r \leq 3$. Then by Fermat's little Theorem, we have

$$b^n \equiv b^{4k + r} \equiv (b^4)^kb^r \equiv 1^kb^r \equiv b^r\pmod{5} \text{for any integer b}.$$

And I guess the author applied FLT for $b^4 \equiv 1 \pmod{5}$
But we don't know that $b$ divides 5 or not, how could the statement above be true? Any idea?

Thanks,
Chan

Answer

The exact statement of what you wrote is slightly incorrect. If $r=0$ and $b=5$ then it does not hold.

Most likely the condition $\gcd (b,5)=1$ is missing.