Congruent modulo

Definition:

• Let $a,b\in \mathbb{Z},m\in {\mathbb{N}}^{+}$.
• If $m|(a-b)$, i.e. of there exists $k\in \mathbb{Z}$, such that $a-b=km$
• We say that $a$ and $b$ are congruent modulo m
  • Denoted by $a\equiv b(\mathrm{mod}b)$
• Example: 4 and 9 are congruent modulo 5

Theorem:

1. Let $a$ and $b$ be integers, and let $m$ be a positive integer.
   • Then $a\equiv b(\mathrm{mod}m)$ if and only if $(a\mathrm{mod}m)=(b\mathrm{mod}m)$theorem
2. Let $m$ be a positive integer.
   • The integers a and b are congruent modulo m if and only if there is an integer $k$ such that $a=b+km$.theorem
3. If $a\equiv b(\mathrm{mod}m)$ and $c=d(\mathrm{mod}m)$ then:theorem
   1. $a+c\equiv b+c(\mathrm{mod}m)$
   2. $ac\equiv bd(\mathrm{mod}m)$
4. Then $a\equiv b(\mathrm{mod}m)$ if and only if $(a-b)=km$theorem