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Algebra and Number Theory
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Elementary Number Theory
Divisibility
editDefinition 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
- If and , then . ▶ □
- If and , then .
- If and , then .
- 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 .
▶