Modular Arithmetic: using congruence to find remainder How do I use the fact that if $a = b \pmod n$ and $c = d\pmod n$ then $ac = bd\pmod n$ to find the remainder when $3^{11}$ is divided by $7$?

$3^{11} \equiv (3^2)^5\cdot 3 \equiv 2^5 \cdot 3 \equiv 96 \equiv 5 \pmod 7$
Of course, it's actually easier to employ Fermat's Little Theorem (if you're permitted) like so:
$3^{11} \equiv 3^6 \cdot 3^5 \equiv 1 \cdot (3^2)^2 \cdot 3 \equiv 2^2 \cdot 3 \equiv 12 \equiv 5 \pmod 7$
