document.write( "Question 305377: 324 is multiplied by a positive integer and the answer is both a perfect square and a perfect cube. What is the number used in the mulitiplication? \n" ); document.write( "

Algebra.Com's Answer #218635 by Edwin McCravy(20056) $\"\"$ $\"About$

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324 is multiplied by a positive integer and the answer is both a perfect square and a perfect cube. What is the number used in the mulitiplication?
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document.write( "A number which is both a perfect square and a perfect cube is a\r\n" );
document.write( "perfect sixth power:\r\n" );
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document.write( "Any perfect 6th power is a product of perfect 6th powers.  So to make a \r\n" );
document.write( "perfect sixth power out of 324 by multiplying it by something,\r\n" );
document.write( "we have to multiply it by 2 enough times to make it contain a factor\r\n" );
document.write( ", and we have to also multiply it by 3 enough times to make it \r\n" );
document.write( "contain \r\n" );
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document.write( "So we have to multiply it by 2 four times, which amounts to multiplying\r\n" );
document.write( "it by , and we also have to multiply it by 3 two times, which \r\n" );
document.write( "amounts to multiplying it by .\r\n" );
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document.write( "So we multiply 324 by \r\n" );
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document.write( "When we multiply 324 by 144 we get 46656 which is both  as well as .\r\n" );
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document.write( "So the smallest possible answer is 144.  \r\n" );
document.write( "  \r\n" );
document.write( "There are other answers too, but 144 is the smallest.\r\n" );
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document.write( "The next larger integer we could have used is to multiply 324 by 2 enough \r\n" );
document.write( "times to make it a perfect twelfth power, which would be to multuiply it by\r\n" );
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document.write( "When we multiply 324 by 9216 we get 2985984 which is both  as well as .\r\n" );
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document.write( "There are actually an infinite number of correct answers, but I think\r\n" );
document.write( "your teacher meant just the smallest answer, which is 144.\r\n" );
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document.write( "Edwin

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