Are the following congruences true?

Question

For $a,b,c,d,m \in\Bbb Z$

If $a\equiv b\mod m$ and $c\equiv d\mod m$

Are the following two statements true?

$a +c \equiv b+d\mod m$

$a*c\equiv b*d\mod m$

The books I come across only list the above for $c=d$, so I wanted to know if these were true as well for $c\neq d$.

Any help is appreciated.

Answer 1

HINT:

$$a+c-(b+d)=a-b+c-d$$

$$ac-bd=a(c-d)+d(a-b)=c(a-b)+b(c-d)$$