modular reductions

Question

How do I use modular reductions to compute this: 11·18·2322·13·19(mod 7)?

I know the answer is 6 through 11=4mod7, 18=4mod7, 2322=5mod7, 13= -1mod7, and 19= 5mod7.

I'm curious as to how you can simplify 2322=5mod7 without using a calculator.

Answer 1

It gets a little better if you use $$ 1001 = 7 \cdot 11 \cdot 13 $$ so $$ 2322 \equiv 320 \pmod 7 $$ Next I suggest $350 = 7 \cdot 50,$ with $49 = 7 \cdot 7,$ so $350 - 49 = 301 \; $ is a multiple of $7.$ Thus $$ 320 \equiv 19 \pmod 7 $$ Then $$ 19 \equiv 5 \pmod 7 $$