# Divisibility

Pronunciation: /dɪˌvɪz.əˈbɪl.ɪ.ti/ Explain

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. In formulas, divisibility 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.

### 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 b | c then a | c.
• If a | b and b | a then a = b or a = -b.

1. McAdams, David E.. All Math Words Dictionary, divisibility. 2nd Classroom edition 20150108-4799968. pg 65. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. Gilbert, Jimmie; and Gilbert Linda. Elements of Modern Algebra. 6th edition. pp 70-75. Thomson, Brooks/Cole. 2005. Last Accessed 7/3/2018. Buy the book

McAdams, David E. Divisibility. 4/20/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/d/divisibility.html.

### Revision History

4/20/2019: Updated equations and expressions to the new format (McAdams, David E.)
3/11/2019: Added clarifying wording. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
7/4/2018: Removed broken links, updated license, implemented new markup, implemented new Geogebra protocol. (McAdams, David E.)
5/5/2011: Initial version. (McAdams, David E.)

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