prove that $3^{11}+11^{13}+13^{17}-1$ and $3^{11}\cdot 11^{13}\cdot 13^{17}-1$ are divisible by 35

Question

How can I prove that $3^{11}+11^{13}+13^{17}-1$ and $3^{11}\cdot 11^{13}\cdot 13^{17}-1$ are divisible by 35?

Thank You very much!

Answer 1

HINT:

As $35=5\cdot7$ with $(5,7)=1$

$3^2\equiv-1\pmod5\implies3^{11}=3(3^2)^5\equiv3(-1)^5\equiv2$

$11\equiv1\pmod5\implies11^{13}\equiv1$

$(13,5)=1$ and $17\equiv1\pmod{\phi(5)},13^{17}\equiv13^1\pmod5\equiv3$

Check for $\pmod7$

But what is more important for me if how $3^{11},11^{13},13^{17}$ are chosen?