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)

You’re using the binary “mod” operator, which may be defined this way: $a\bmod m$ is the number $\bar a$ satisfying the conditions that $0\le\bar a<m$ and $a-\bar a$ is a multiple of $m$. In other words, $a\bmod m$ is the remainder when $a$ is divided by $m$ using Euclidean division.
As you’ve undoubtedly noticed, it may be that $(a\bmod m)+(b\bmod m)$ may be unequal to $(a+b)\bmod m$, as in the case $a=4$, $b=5$, and $m=7$. But notice that their difference is still a multiple of $7$.
Mathematicians say, when $A$ and $B$ have a difference that’s divisible by $m$, that $A\equiv B\pmod m$. I think that this notation has slipped into the formulation of your problem, which seems to be asking you to show that even when $(a\bmod m)+(b\bmod m)\ne(a+b)\bmod m$, the two numbers still have a difference divisible by $m$.
In the notation I introduced for $\bar a$, you’re being asked to show that $\bar a+\bar b$ and $\overline{a+b}$ are numbers whose difference is divisible by $m$. But that’s easy enough: $a-\bar a$, $b-\bar b$ and $a+b-\overline{a+b}$ all are multiples of $m$. That is, $a-\bar a=rm$, $b-\bar b=sm$, $a+b-\overline{a+b}=tm$. Pile them all together in the right combination, and you get $\bar a+\bar b-(\overline{a+b})=(t-r-s)m$. I’ll let you do the other one in (almost) the same way.
