Prove that $2^{15}-1$ is divided by $11\cdot31\cdot61$? I have to prove that $2^{15}-1$ is divided by $11\cdot31\cdot61$. 
I have proven using congruencies that $2^{15}-1$ is divided by $31$. However we have
$$2^5\equiv 10 \mod{11}$$
$$2^{15}\equiv 10^3=1000\equiv 10 \pmod {11}$$
Therefore
$$2^{15}-1\equiv 9 \pmod{11}.$$
So it is impossible to prove!!

You are correct, there is a mistake in the problem. Here are two useful prime factorizations:
$$
\begin{align*}
2^{15}-1&=32767=7\cdot 31\cdot 151\\
2^{15}+1&=32769=3^2\cdot 11\cdot 331
\end{align*}
$$
