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divisibility (Definition)

Given integers $ a$ and $ b$, then we say $ a$ divides $ b$ if and only if there is some $ q \in \mathbb{Z}$ such that $ b=qa$.

There are many other ways in common use to express this relationship:

  • $ a\mid b$ (read “$ a$ divides $ b$”).
  • $ b$ is divisible by $ a$.
  • $ a$ is a factor of $ b$.
  • $ a$ is a divisor of $ b$.
  • $ b$ is a multiple of $ a$.

The notion of divisibility can apply to other rings (e.g., polynomials).



"divisibility" is owned by mathcam. [ full author list (2) | owner history (1) ]
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See Also: least common multiple, example of gcd, $\tau$ function, exactly divides, divisor sum of an arithmetic function, strict divisibility, fundamental theorem of arithmetic

Other names:  divides, divisor, factor, multiple

Attachments:
divisibility in rings (Definition) by pahio
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Cross-references: polynomials, rings, divisible, integers
There are 332 references to this entry.

This is version 7 of divisibility, born on 2001-11-16, modified 2007-09-11.
Object id is 923, canonical name is Divisibility.
Accessed 20735 times total.

Classification:
AMS MSC: 11A51 (Number theory :: Elementary number theory :: Factorization; primality)
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