Given that a= b mod n and c= d mod n , to prove a+c = b+d mod n .

Now a = b mod n means a-b is divisible by n, Like wise c-d is divisible by n.

Add both the congruences a-b + c-d will be divisible by n

Or, (a+c) –(b+d) is divisible by n

In other words, a+c = b+d mod n Proved