Algebra.Com's Answer #623326 by cparks1000000(5)![\"\"](\"/images/stars/iconNewId_16x16.gif\")  
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document.write( "Assume  is rational. Then there exists integers  and  such that  . Then by Zorn's lemma, there exists a greatest common divisor of and , say it is  . Set  such that  and  such that  . Then by definition of the rationals,  . But then  divides  . Then by definition of the rationals,  . This is true since if we assume that divides ; then we can use the unique factorization theorem to conclude that  where all of the  are prime and none of them are 7. Thus divides  which is a contradiction since it obviously has no factors which are since is prime. Thus we have that divides  . Since divides , there exists  such that  , thus  . Thus by similar logic to the above, we have  divides . Since  divides we can use the prime factorization theorem to once again conclude that divides . But then and must have a common factor. This is a contraction. Thus  is irrational. \n" );
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