Mathematics

Theorem:   For given any integers $a $ and $b$, $a \equiv b ~(mod ~n) $ if and only if $a$ and $b$ leave the same nonnegative remainder when  divided by $n$. Proof: Firstly, we suppose that  $a \equiv b ~(mod ~n)$ and we will show that $a$ and $b$ leave the same nonnegative remainder when divided by $n$. Since  $a \equiv b ~(mod ~n)$, we have $a = b + k \cdot n$ for some integer $k$. When we divide $b$ by $n$,  $ b$ leaves a certain remainder say $r$; that is,  $b = q \cdot n + r$, where $0 \leq r < n$. We put $b = q \cdot n + r$ in $a = b + k \cdot n$.  Therefore, $a =  b + k \cdot n = (q \cdot n + r) + k \cdot n = (q + k) \cdot n + r$ that is  $a$ has the same remainder as $b$, when we divide $a$ by $n$.  Conversely, suppose that $a$ and $b$ leave the same nonnegative remainder when divided by $n$. So  we can write $a = q_{1} \cdot n + r$ and $b = q_2 \cdot n + r$, with the  same remainder $0 \leq r < n$. Then $a...

  Definition:  The set of $n$ integers $0, ~1, ~2, ... ,~ n - 1$ is called the set of least nonnegative residues  modulo $n$. In general, a collection of $n$ integers $a_{1},~ a_2, \cdots ,~ a_n$ is said to form a complete set of residues or a complete system of residues modulo $n$ if every integer is congruent modulo $n$ to one and only one of the $a_k$ where $1\leq k \leq n$. We can observe that The set of $5$ integers $0, ~1, ~2, ~3,~ 4$ is  the set of least nonnegative residues modulo $5$ and a collection of $5$ integers $10, ~21, ~37, ~43, ~59$  form a complete set of residues (or  a complete system of residues ) modulo $5$. Observe that any $n$ integers form a complete set of residues modulo $n$ if and only if no two of the integers are congruent modulo $n$. Remark:  Note that for given integer $n > 0$, negative integer can  form a complete set of residues (or  a complete system of residues ) modulo $n$. For example $59 \e...