document.write( "Question 381701: hello all, \r
\n" ); document.write( "\n" ); document.write( "I am taking grad level complex algebra and i will appreciate if anyone can help me with the following problem:\r
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Algebra.Com's Answer #270793 by Edwin McCravy(20067) $\"\"$ $\"About$

You can put this solution on YOUR website!

First a little background, which you may already be familiar with.\r\n" );
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document.write( "[Since no complex integer can be said to be \"greater\" than another, nor\r\n" );
document.write( "the \"greatest\" in a set, one might think it should be meaningless to speak of\r\n" );
document.write( "a \"greatest\" common divisor.  However not all common divisors are divisible\r\n" );
document.write( "by all other common divisors.  For instance ±1, ±i and 1+i are all common\r\n" );
document.write( "divisors of 2+2i and 3+3i. However ±1, ±i are not divisible by 1+i, athough\r\n" );
document.write( "1+i is divisble by them.  We reserve the term \"greatest\" for a common divisor\r\n" );
document.write( "that is divisible by ALL common divisors.  \"Greatest\" is still a misnomer,\r\n" );
document.write( "since none are really \"greater\" than any of the others, but we still use the\r\n" );
document.write( "term \"GCD\" to refer to any common divisor that is divisible by ALL common\r\n" );
document.write( "divisors.]  \r\n" );
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document.write( "Since there are 4 unit complex integers, ±1, ±i, there are 4 associate\r\n" );
document.write( "GCD's of any two complex integers.  That is, if a+bi is a GCD of p+qi\r\n" );
document.write( "and r+si then all four of these are also GCDs of p+qi and r+si.\r\n" );
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document.write( " 1(a+bi) =  a+bi\r\n" );
document.write( "-1(a+bi) = -a-bi \r\n" );
document.write( " i(a+bi) =  ai+bi² = ai+b(-1) = ai-b = -b+ai\r\n" );
document.write( "-i(a+bi) = -ai-bi² = -ai-b(-1) = -ai+b = b-ai \r\n" );
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document.write( "To use the Euclidean algorithm, we begin by dividing one by the other until\r\n" );
document.write( "we get a remainder of 0.\r\n" );
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document.write( "What do you know!  We got a complex integer on the first division.\r\n" );
document.write( "The remainder is therefore 0, since we got a complex integer, in fact,\r\n" );
document.write( "the complex unit i, so the set of four associate GCDs are 1,-1, i, and -i\r\n" );
document.write( "times the last divisor used, which was 4-3i, and they are, as we saw above:\r\n" );
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document.write( "{4-3i, -4+3i, 3+4i, -3-4i} \r\n" );
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document.write( "Those are the answers.\r\n" );
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document.write( "[Notice that there are other common divisors of 3+4i and 4-3i, and not\r\n" );
document.write( "just ±1 and ±i, either.  For 2+i, -2-i, 1-2i, and -1+2i are also common\r\n" );
document.write( "divisors of 3+4i and 4-3i as well.  However, none of these are divisible by\r\n" );
document.write( "any of the four associate GCDs, so they are not GCDs.  However the four\r\n" );
document.write( "associate GCDs are divisible by them.]   \r\n" );
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document.write( "Edwin

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