TITLE: Let $n=2047$. Using the fact that $3^{88} \equiv 1\pmod {n}, 3^{55} \equiv 1565\pmod {n}.$ Show that n is a composite number.
QUESTION [6 upvotes]: Let $n=2047$. 
Using the fact that $$3^{88} \equiv 1\pmod {n}, 3^{55} \equiv 1565\pmod {n}.$$ Show that n is a composite number.
This is a question on a past exam that I find difficulty to answer. Can anyone help?

REPLY [2 votes]: An answer from group theory: if $\;n=2047\;$ is prime, then $\;\Bbb F_n^*\;$ is a (cyclic) group, so by Lagrange's theorem
$$3^{88}=1\pmod n\implies 88\mid |\Bbb F_n^*|=n-1=2046$$
But, of course, it is false that $\;88\mid 2046\;$...for example, it's easy to see that $\;2046=2\pmod 4\;$ , whereas $\;88=0\pmod 4\;$ .