Using the definition of congruence and Fermat's Little Theorem

Question

I'm currently working on this problem:

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Use the definition of congruence and Fermat's Little Theorem to show that if gcd$(b,561) = 1$, then $$b^{560} \equiv \begin{equation} \begin{cases} 1 &\text{ mod } 3,\\ 1 &\text{ mod } 11,\\ 1 &\text{ mod } 17.\\ \end{cases} \end{equation}$$

Conclude that, if gcd$(b,561)=1$, then $b^{560}\equiv1\text{ mod }561$ and so $561$ is a Carmichael number. Hint: What does it mean that $b^{561}\equiv 1\text{ mod }561?$

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So I'm guessing that it's significant that 3,11, and 17 all divide 561. Fermat's little theorem doesn't, by itself, provide for something like $b^{p-1} \equiv 1\text{ mod }q$ where q and p are different, so I'm guessing this is where I need to use congruence to prove that 561 is congruent to 3, 11, and 17?

Answer 1

Because $561=3\cdot 11 \cdot 17$, where $3,11,17$ all are primes, you will have things like $$b^2 \equiv 1 \pmod{3} \rightarrow (b^2)^{280} \equiv 1^{280} \pmod{3}$$ $$b^{10} \equiv 1 \pmod{11} \rightarrow (b^{10})^{56} \equiv 1^{56} \pmod{11}$$ $$b^{16} \equiv 1 \pmod{17} \rightarrow (b^{16})^{35} \equiv 1^{35} \pmod{17}$$ or $$b^{560} \equiv 1 \pmod{3}$$ $$b^{560} \equiv 1 \pmod{11}$$ $$b^{560} \equiv 1 \pmod{17}$$ or $$b^{560} - 1 =3q_1$$ $$b^{560} - 1 =11q_2$$ $$b^{560} - 1 =17q_3$$ and applying Euclid's lemma two times for $3q_1=11q_2=17q_3$ $$b^{560} - 1 =561q$$ or $$b^{560} \equiv 1 \pmod{561}$$