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Homedivisibility by prime number

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# divisibility by prime number

###### Theorem.

Let $a$ and $b$ be integers and $p$ any prime number. Then we have:

$\displaystyle p\mid ab\quad\Leftrightarrow\quad p\mid a\;\lor\;p\mid b$ | (1) |

Proof. Suppose that $p\mid ab$. Then either $p\mid a$ or $p\nmid a$. In the latter case we have $\gcd(a,\,p)=1$, and therefore the corollary of Bézout’s lemma gives the result $p\mid b$. Conversely, if $p\mid a$ or $p\mid b$, then for example $a=mp$ for some integer $m$; this implies that $ab=mb\cdot p$, i.e. $p\mid ab$.

Remark 1. The theorem means, that if a product is divisible by a prime number, then at least one of the factor is divisibe by the prime number. Also conversely.

Remark 2. The condition (1) is expressed in terms of principal ideals as

$\displaystyle(ab)\subseteq(p)\quad\Leftrightarrow\quad(a)\subseteq(p)\,\lor\,(% b)\subseteq(p).$ | (2) |

Here, $(p)$ is a prime ideal of $\mathbb{Z}$.

Related:

PrimeElement, DivisibilityInRings, EulerPhiAtAProduct, RepresentantsOfQuadraticResidues

Synonym:

divisibility by prime

Type of Math Object:

Theorem

Major Section:

Reference

Parent:

Groups audience:

## Mathematics Subject Classification

11A05*no label found*

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