Question

# Prove that if a -= b(mod n) and c -= d(mod n), then if a-c -= b-d(mod n)

Congruence
Prove that if $$\displaystyle{a}\equiv{b}{\left(\text{mod}{n}\right)}{\quad\text{and}\quad}{c}\equiv{d}{\left(\text{mod}{n}\right)}$$, then if $$\displaystyle{a}-{c}\equiv{b}-{d}{\left(\text{mod}{n}\right)}$$

2021-02-25
By the definition of Congruence, there are integers s and t such that,
a-b = sn...(1)
and
c-d=tn...(2)
(1)-(2)
(a-b)-(c-d)=sn-tn
(a-b)-(c-d)=n(s-t)
Both s and t are integers so s−t is also integer
put s−t=u
(a-b)-(c-d)=nu
By defination of Congruence
$$\displaystyle{\left({a}-{b}\right)}\equiv{\left({c}-{d}\right)}{\left(\text{mod}{n}\right)}$$
Hence Proved,
$$\displaystyle{\left({a}-{c}\right)}\equiv{\left({c}-{d}\right)}{\left(\text{mod}{n}\right)}$$