document.write( "Question 27106: show that the only common factor of any two consecutive positive odd integers is 1. \n" ); document.write( "

Algebra.Com's Answer #14736 by kev82(151) $\"\"$ $\"About$

You can put this solution on YOUR website!
Hi,\r
\n" ); document.write( "\n" ); document.write( "The fundamental theorem of arithmatic tells us that any posotive intrger can be factorised as a product of primes. Using this lets write the consecutive odd integers as

and

\r
\n" ); document.write( "\n" ); document.write( "Arbitrarily choose a prime factor of

called

and consider

. (If

is a common factor of

and

then

will be an integer.)\r
\n" ); document.write( "\n" ); document.write( "

\r
\n" ); document.write( "\n" ); document.write( "For this to be an integer,

must be an integer. This is only true if

. But

and

are both odd, so can't have a factor of two, thus the only common factor is one.\r
\n" ); document.write( "\n" ); document.write( "Hope that helps,\r
\n" ); document.write( "\n" ); document.write( "Kev \n" ); document.write( "

\n" );