Algebra and Number Theory/Elementary Number Theory

DivisibilityEdit

Definition 1: (divides, divisor, multiple)

Let  , with  . We say that "  divides  " or that "  is a multiple of  ", if there exists some   such that  .

We write this as  .

Proposition 1: (some elementary properties of division)

Let   be integers. Then

  1. If   and  , then  . ▶  
  2. If   and  , then  .
  3. If   and  , then  .
  4. If   and  , then  . ▶  

Examples:   because  . However  : if it did, would also divide   (by Proposition 1, point 3), which is impossible (Proposition 1, point 1). Similarly,  .

Proposition 2: (division algorithm)

Let  , with  . Then  , for some  , with  .