Reduction Modulo N vs. Euclidean Division by N

Modular Arithmetic : Introduction

What is modular arithmetic? We'll get to that in a minute - first start with this.

Correct!

Another question:

Notice that the remainders are the same when we divide $29$ and $2014$ by $5$.

It is sort of lengthy to write out this sentence. Modular arithmetic gives us the notation to write this sort of statement, which is called an equivalence. In modular arithmetic notation, we could write $2014 \equiv 29 \pmod{5}$.

One last preliminary question before we give a formal definition:

Now, how are these questions related to modulo arithmetic? Pick the correct definition :

Two integers $a$ and $b$ are congruent modulo $m$ (where $m$ is a positive integer) if and only if $m \mid a - b$.

In other words, if two numbers $a, b$ leave the same remainder when divided by $m$, they are congruent. This is usually denoted as $a \equiv b \pmod{m}$ or sometimes $a \equiv_{m} b$.

We'll look at the properties of modular arithmetic in the next part.

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