document.write( "Question 25736: How do i show for intergers a, b and k that gcd(a,b)= gcd(a,b+ka).\r
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Algebra.Com's Answer #14667 by venugopalramana(3286) $\"\"$ $\"About$

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How do i show for intergers a, b and k that gcd(a,b)= gcd(a,b+ka).
\n" ); document.write( "LET GCD(A,B+KA)=D
\n" ); document.write( "HENCE D=AX+(B+KA)Y...WHERE X AND Y ARE INTEGERS.
\n" ); document.write( "D=A(X+KY)+BY..SINCE K IS AN INTEGER,X+KY IS AN INTEGER.
\n" ); document.write( "SO D IS THE GCD OR MULTIPLE OF GCD OF A AND B.
\n" ); document.write( "IF WE ASSUME GCD(A,B)=G...
\n" ); document.write( "THEN G|D..........................................I
\n" ); document.write( "SIMILARLY,SINCE GCD(A,B)=G WE GET
\n" ); document.write( "G=PA+QB...WHERE P AND Q ARE INTEGERS
\n" ); document.write( "G=PA-KAQ+QB+KAQ=A(P-KQ)+Q(B+KA).....(P-KQ) AND Q ARE INTEGERS.
\n" ); document.write( "SO G IS THE GCD OR MULTIPLE OF GCD OF A AND B+KA
\n" ); document.write( "HENCE D|G.......................................II
\n" ); document.write( "I AND II CAN BE TRUE ONLY IF D=G...PROVED \n" ); document.write( "

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