How can I show that $4^{1536} - 9^{4824}$ can be divided by $35$ without remainder? How can I show that $4^{1536} - 9^{4824}$ can be divided by $35$ without remainder?
I'm not even sure how to begin solving this, any hints are welcomed!
$$(4^{1536} - 9^{4824}) \pmod{35} = 0$$

Euler's theorem implies, since $\varphi(35)=24$, that
$$4^{24}\equiv 9^{24}\equiv 1\pmod {35}$$
Since $1536$ and $4824$ are multiples of $24$, the conclusion follows:
$$4^{1536}-9^{4824}=(4^{24})^{64}-(9^{24})^{201}\equiv1^{64}-1^{201}\equiv 0\pmod{35}$$
