Primes and Divisibility

• Set of integers is denoted as $\mathbb{Z}$
• for $a, b \in \mathbb{Z}$,
  • a divides b is denoted as $a \mid b$
  • a does no divides b is denoted as $a \nmid b$
• if $a \mid b$ then
  • $a$ is a divisor of $b$
  • if $a \notin {1, b}$ then $a$ is a non trivial divisor or a factor of $b$
• a positive integer $p \gt 1$ is prime if it has not factors.
• the greatest common divisor of two integers $a$ and $b$ is denoted as $gcd(a, b)$ and is the largest $c$ such that $c \mid a$ and $c \mid b$
• if $p$ is prime then $gcd(a, p) = 1 \text{ or } p$
• if $gcd(a, b) = 1$ then $a, b$ are said to be co-prime

Modular Arithmetic