Hi I do not understand the answers to these 2 questions:
1. Can you work out what the remainder is when 2^1000 is divided by 7?
As 1000 = 333 × 3 + 1 then 2^1000 leaves remainder 2 when divided by 7.
2. Expand the brackets of (x − 1)(x − 7) and (x − 3)(x − 5) in mod 8 arithmetic.
(x − 1)(x − 7) = x^2 − 8x + 7 = x^2 − 1 (mod 8);
(x − 3)(x − 5) = x^2 − 8x + 15 = x^2 − 1 (mod 8).
Why is there a "-1", we dont take remainders to be negative?
I'm new to modular arithmetic, just started it right now, so apologies if its a stupid qustion.