Divisibility

Description:

• Let $a,b\in Z$ with $a\ne 0$. We say that $a$ divides $b$, denoted by $a|b$, if there exists some $k\in Z$ such that $b=ak$
• In equation, $a=dq+r$
  • $a$ is called the dividend
  • $d$ is called the divisor
  • $q$ is the quotient
    • $q=a\text{ div }d$
  • $r$ is the remainder
    • $r=a\mathrm{mod}d$

Theorems:

• Let $a,b,c\in \mathbb{Z}$:
  1. If $a|b$ and $a|c$ then $a|(b+c)$theorem
  2. If $a|b$ then $a|bc$theorem
  3. If $a|b$ and $b|c$ then $a|c$ (i.e, transitivity)theorem
• Division algorithm:theorem
  • For any $a\in \mathbb{Z}$ and $d\in {\mathbb{N}}^{+}$, there exist unique $q,r\in \mathbb{Z}$ such that:
    • $a=dq+r$
    • and $0\le r<d$