$a^{37} \equiv a \pmod {1729}$ Use Euler theorem to show that
$$a^{37} \equiv a \pmod {1729}$$ is true for any integer $a$.
I am a beginner in Number theory and stuck in this problem. Show me the right direction.

By Chinese Remainder Theorem $R=\mathbb{Z}_{1729}\cong\mathbb{Z}_7\times\mathbb{Z}_{13}\times\mathbb{Z}_{19}$. Write the elements of $R$ in three coordinates, then you can easily see the result.
