Congruences - proof problems

Question

1) State what is meant by $a\equiv b \pmod n$.

2) Suppose that $a\equiv b \pmod n$ and $c\equiv d \pmod n$. Prove that

i) $a+c\equiv b+d \pmod n$

ii) $ac\equiv bd \pmod n$

For question 1, I would state that $n|(a-b)$

For part ii, I would simply use algebraic manipulation to get the desired result but i don't think I'm proving it.

Answer 1

1) is perfect.

For 2i) use your definition: if $n|a-b$ and $n|c-d$ then $n|(a-b) + (c-d) = (a+c)-(b+d)$.

For 2ii) you can very well try the same kind of proof. Can you try it yourself?

Answer 2

Let $a = k_1n + b$ and $c = k_2n + d$, $k_1, k_2 \in \Bbb Z$. Then $$ a+c = k_1n + b + k_2n + d = (k_1+k_2)n + (b+d)\\ ac = (k_1n+b)(k_2n+d) = (k_1k_2n+k_1d+k_2b)n+bd$$

Answer 3

If $m$ divides $a,b;m$ must divide $ax+by$ where $x,y$ are arbitrary integers

For the first, $$a+c-(b+d)=(a-b)+(c-d)$$

For the second, $$ac-bd=a(c-d)+d(a-b)$$