document.write( "Question 26947: PLEASE NEED HELP
\n" ); document.write( "show that the set of all integers of form 36a-90b+72c forms an ideal in Z(intergers) and find the generator for the ideal.
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Algebra.Com's Answer #15095 by venugopalramana(3286)\"\" \"About 
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LET US DENOTE THE SET OF ALL INTEGERS OF FOR
\n" ); document.write( "36K-90L+72M BY S.
\n" ); document.write( "K,L,M ARE INTEGERS
\n" ); document.write( "TO SHOW THAT S IS AN IDEAL OF Z.
\n" ); document.write( "LET S1 AND S2 BE 2 ELEMENTS OF S SO THAT
\n" ); document.write( "S1=36K1-90L1+72M1 AND S2=36K2-90L2+72M2
\n" ); document.write( "THEN S1-S2=36(K1-K2)-90(L1-L2)+72(M1-M2)
\n" ); document.write( "SINCE K1-K2,L1-L2,M1-M2 ARE INTEGERS ,S1-S2 IS IN THE FORM
\n" ); document.write( "S=36K-90L+72M
\n" ); document.write( "SO S IS A SUB GROUP OF Z UNDER ADDITION.
\n" ); document.write( "NOW LET Z1 BE ANY ELEMENT OF Z ; AND S1 AS ABOVE AN ELEMENT OF S.THEN
\n" ); document.write( "WE HAVE Z1S1=Z1*36K1-Z1*90L1+Z1*72M1…
\n" ); document.write( "SINCE MULTIPLICATION OF INTEGERS IS ASSOCIATIVE AND COMMUTATIVE IN Z,WE GET
\n" ); document.write( "Z1S1=S1Z1=36(K1Z1)-90(L1Z1)+72(M1Z1),WHICH BEING OF THE FORM S
\n" ); document.write( "ARE ELEMENTS OF S.
\n" ); document.write( "HENCE S IS LEFT AND RIGHT IDEAL OF Z ,THAT IS AN IDEAL OF Z..
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\n" ); document.write( "S=36K-90L+72M
\n" ); document.write( "THIS MEANS S IS THE GCD OF 36,90,72 OR MULTIPLE OF THEIR GCD.
\n" ); document.write( "SO GCD OF 36,72,90 IS A GENERATOR OF THIS IDEAL.THAT IS 18 IS A GENERATOR. \r
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