document.write( "Question 831380: The LCM of 6, 12, and n is 660. Find all the possible values of n. \n" ); document.write( "

Algebra.Com's Answer #501363 by KMST(5328) $\"\"$ $\"About$

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$\"660=10%2A66=10%2A6%2A11=5%2A2%2A2%2A3%2A11=2%5E2%2A3%2A5%2A11\"$
\n" ); document.write( "so the prime factorization is $\"660=2%5E2%2A3%2A5%2A11\"$
\n" ); document.write( "Since the prime factorization for $\"6\"$ and $\"12\"$ are
\n" ); document.write( " $\"6=2%2A3\"$ and $\"12=2%5E2%2A3\"$ ,
\n" ); document.write( "the factors $\"5\"$ and $\"11\"$ must come from $\"n\"$ .
\n" ); document.write( "There could be other factors of $\"n\"$ included in $\"2%5E2%2A3%2A5%2A11\"$ .
\n" ); document.write( "The prime factorization of $\"n\"$ could have
\n" ); document.write( " $\"2\"$ with exponents 0, 1, or 2,
\n" ); document.write( "and/or $\"3\"$ with exponents 0 or 1,
\n" ); document.write( "in addition to $\"5\"$ and $\"11\"$ .
\n" ); document.write( "That gives us $\"3%2A2=6\"$ possible values for $\"n\"$ :
\n" ); document.write( " $\"2%5E0%2A3%5E0%2A5%2A11=highlight%2855%29\"$
\n" ); document.write( " $\"2%5E0%2A3%5E1%2A5%2A11=highlight%28165%29\"$
\n" ); document.write( " $\"2%5E1%2A3%5E0%2A5%2A11=highlight%28110%29\"$
\n" ); document.write( " $\"2%5E1%2A3%5E1%2A5%2A11=highlight%28330%29\"$
\n" ); document.write( " $\"2%5E2%2A3%5E0%2A5%2A11=highlight%28220%29\"$
\n" ); document.write( " $\"2%5E2%2A3%5E1%2A5%2A11=highlight%28660%29\"$ \n" ); document.write( "

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