divisibility by prime number


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

pabpapb (1)

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

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 of principal idealsMathworldPlanetmathPlanetmathPlanetmath as

(ab)(p)(a)(p)(b)(p). (2)

Here, (p) is a prime idealMathworldPlanetmathPlanetmath of .

Title divisibility by prime number
Canonical name DivisibilityByPrimeNumber
Date of creation 2013-03-22 14:48:18
Last modified on 2013-03-22 14:48:18
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 18
Author pahio (2872)
Entry type Theorem
Classification msc 11A05
Synonym divisibility by prime
Related topic PrimeElement
Related topic DivisibilityInRings
Related topic EulerPhiAtAProduct
Related topic RepresentantsOfQuadraticResidues