Divisibility

Pronunciation: /dɪˌvɪz əˈbɪl ɪ ti/ ?

An integer i is divisible by another integer j if, when i is divided by j, there is no remainder. For example, 12 is divisible by 3 since 12÷3 = 4 with a remainder of 0. This is written with a vertical bar |. For example, write 3|12 and say "3 divides 12". If j divides i, j is also a factor of i.

Mathematical definition

Properties of divisors

For integers a, b, c, m, n, p and q:

  • If a|b and a|c then a|(b+c) and a|(mb+mc).
    If a|b then there exists an integer p such that a·p = b. Similarly, there exists an integer q such that a·q = c.
  • If a|b and a|c then a|c.
  • If a|b and b|a then a = b or a = -b.

References

  1. Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra, 6th edition, pp 70-75. Thomson, Brooks/Cole, 2005.

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Divisibility. 2010-02-24. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/d/divisibility.html.

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2010-02-24: Authored (McAdams, David.)

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