I'm struggling a little with a question concerning power towers.
I have this number $2018^{\large {2017}^{\Large 16050464}}\!$ and I want to find the remainder when it is divided by 1001.
I have managed to reduce it down to $16^{\large {2017}^{\Large 224}}$ but I can't get any further past this point. I feel like I need to reduce $2017^{224}$ to an integer but I am unsure of how to do this.
Just a side note: I have to do it without the Chinese remainder theorem.
Any help is welcome, thanks.
$16^{\large 3}\!\equiv 1$ mod $7\,\&\,13;\,$ $16^{\large 5}\!\equiv 1\pmod{\!11}\,$ so $\,16^{\large\color{#0a0}{15}}\!\equiv 1\,$ mod $7,11,13$ so also mod their lcm $=1001$.
Thus $\bmod 1001\!:\ 2018^{\large 2017^{\Large 4N}}\!\!\!\equiv 16^{\large 2017^{\Large 4N}\!\!\bmod\color{#0a0}{15}}\!\!\equiv 16^{\large \color{#c00}7^{\Large\color{#c00} 4N}\!\!\bmod\color{#0a0}{15}}\!\!\equiv 16^{\large\color{#c00} 1^{\Large N}}\!\!\equiv 16$
with the expt calculation: $\bmod\color{#0a0}{15}\!:\,\ 2017\equiv 7,\ $ and $\ \color{#c00}7^{\large\color{#c00}4}\! \equiv 4^{\large 2}\!\equiv\color{#c00} 1$