Algebra and Number Theory/Printable version


Algebra and Number Theory

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Sets

Set OperationsEdit

Special SetsEdit

Functions and Binary OperationsEdit

Equivalence RelationsEdit



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 .