Help needed with modulus addition and multiplication proof We have recently started working with modular arithmetic in my discrete mathematics course, and I found two problems in my textbook that I am having trouble with. What are these kinds of proofs called, and what is the usual approach that is undertaken? Lastly, how would you suggest tackling these proofs in particular? Thank you so much in advance!
1.(a mod m) + (b mod m) ≡ (a + b mod m)
2.(a mod m)(b mod m) ≡ (ab mod m)

The idea here is to write $a\mod m$ as $a+k_1m$, where $k_1$ is some constant. Then we can construct a proof quite easily.
For 1, we have $(a+k_1m) + (b + k_2m) = a+b+(k_1+k_2)m \equiv a+b\mod m$
And for 2, we have $(a+k_1m)(b + k_2m) = ab+(bk_1+ak_2+k_1k_2m)m \equiv ab \mod m$
