Mathematics

  Definition: An integer $b$ is said to be divisible by an integer $a \not = 0$, if there exists some integer $c$ such that $ b = a \cdot c$. An integer $b$ is  divisible by an integer $a \not = 0$, is denoted by $a ~|~ b$.  We write $a \nmid b$ to indicate that $b$  is not divisible by $a$. Theorem: For integers $a, ~b, ~c$ the following hold: $1.$ $a ~|~ 0, ~ 1 ~|~ a, ~ a ~|~ a $. $2.$  $ a ~|~ 1 $ if and only if $a =  1$ or $a = -1$. $3.$  If $a ~|~ b$ and $c ~|~ d$, then $a \cdot c ~|~ b \cdot d$. $4.$   If $a ~|~ b$ and $b ~|~ c$, then $a ~|~ c$. $5.$  If $a ~|~ b$ and $b ~|~ a$ if and only if $a =  b$ or $a = -b$. $6.$  If $a ~|~ b$ and $b \not = 0$, then $|a| \leq |b|$. $7.$  If $a ~|~ b$ and $a ~|~ c$, then $a~ |~ (b \cdot x + c \cdot y)$ for arbitrary integers $x$ and $y$. Proof: $1.$ For any integer $a$, we can write $0 = a \cdot 0$, $a = 1 \cdot a$ and $a = a \cdot 1$. Theref...

 Let  $ \mathbb{Z}$ denotes set of all integers, that is, $\mathbb{Z} = \{\dots,~ -4,~-3,~-2,~-1,~0,~1,~2,~3,~4,~ \cdots\}$ and $\mathbb{Z}^{+}$ denotes set of all nonnegative integers, that is, $\mathbb{Z}^{+} = \{0,~1,~2,~3,~4,~ \cdots\}$.    Well-Ordering Principle : Every nonempty set $S$ of a nonnegative integers contains a least element; that is, there exists integer $a \in S$ such that $a \leq b$ for all $b \in S$. If we take set $S = \{8,~ 5, ~90, ~3, ~29, ~57\}$. One can observe that $S$ is nonempty and $S \subseteq \mathbb{Z}^{+}$ and $3 \in S$ such that $3$ is less than or equal to all other elements of $S$. So that $3$ is the least element of $S$.