document.write( "Question 938445: what is the least positive integer that has exactly 9 factors? \n" ); document.write( "

Algebra.Com's Answer #571753 by KMST(5328) $\"\"$ $\"About$

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A number has more or less factors depending on its prime factorization.
\n" ); document.write( "If the prime factorization is
\n" ); document.write( " $\"n=A%5Ea%2AB%5Eb%2A%22....%22Y%5Ey%2AZ%5Ez\"$
\n" ); document.write( "those prime factors can be arranged into products of the form
\n" ); document.write( " $\"A%5E%28alpha%29%2AB%5E%28beta%29%2A%22...%22\"$
\n" ); document.write( "with $\"0%3C=alpha%3C=a\"$ , $\"0%3C=beta%3C=b\"$ , etc
\n" ); document.write( "to form all factors of $\"n\"$
\n" ); document.write( "from $\"1=A%5E0%2AB%5E0%2A%22...%22\"$ to
\n" ); document.write( " $\"n=A%5Ea%2AB%5Eb%2A%22...%22\"$ .
\n" ); document.write( "The number of possible products is
\n" ); document.write( " $\"%28a%2B1%29%2A%28b%2B1%29%2A%22...%22%2A%28y%2B1%29%2A%28z%2B1%29\"$ ,
\n" ); document.write( "with as many such brackets as prime factors,
\n" ); document.write( "(multiplied together if there is more than one),
\n" ); document.write( "and each bracket being 2 or more.
\n" ); document.write( "In this case, $\"9=3%2A3\"$ is the number of factors,
\n" ); document.write( "meaning that there are just 2 prime factors,
\n" ); document.write( "both with 2 as an exponent:
\n" ); document.write( " $\"n=A%5E2%2AB%5E2\"$ with $\"a=b=2\"$ .
\n" ); document.write( "The smallest $\"n\"$ we can make with that formula is
\n" ); document.write( " $\"n=2%5E2%2A3%5E2=4%2A9=36\"$ . \n" ); document.write( "

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