## anonymous 4 years ago 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?

1. 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.

2. anonymous

Thank you.

3. 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 $$540=(3^2.2).3.2.5 = 18.3.2.5$$ 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.