• anonymous
Is this a trick question? "Which pairs of integers \(a\) and \(b\) have greatest common divisor \(18\) and least common multiple \(540\)?" There is a theorem that states that \([a,b](a,b)=ab\), where \([a,b]\) stands for the least common multiple of \(a\) and \(b\), and \((a,b)\) stands for the greatest common divisor of \(a\) and \(b\). So, it follows that\[[a,b](a,b)=ab,\]\[540\cdot18=a\cdot b.\]In other words, \(a=540\) and \(b=18\) would suffice. Am I missing something?
  • Stacey Warren - Expert
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  • jamiebookeater
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  • jagatuba
I don't think you are missing anything. The GCD of 18=18 and the LCM of 540=540, so as far as the problem goes with the figures you are given the theorem holds true.
  • anonymous
Thank you.
  • mathmate
It is probably a trick question, because it asked "Which pairS of integers a and b ..." implies that you need to find ALL the pairs. Your logic is perfectly accurate, so let's continue from there. Break 540 into factors \( 540=(3^2.2).3.2.5 =\) So the pairs of A and B could be taken from the set \(S=\{18,1,3,2,5\} \) such that \(A \cup B = S \ and\ A \cap B = \{18\}\) From this, we can find pairs of a,b such as (18,540), (36,270),(54,180),(90,108), all satisfying the above conditions.

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