document.write( "Question 319260: What is the smallest positive integer n for which 45n is a perfect cube of an integer? \n" ); document.write( "

Algebra.Com's Answer #228589 by jim_thompson5910(35256) $\"\"$ $\"About$

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First, factor 45n into 3*3*5*n. In order for 3*3*5*n to be a perfect cube, each prime factor must come in sets of triples. So we're missing one 3 and two 5 factors which means that n=3*5*5=75\r
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\n" ); document.write( "\n" ); document.write( "So the answer is n=75 making the final number to be 45*75=3375\r
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\n" ); document.write( "\n" ); document.write( "Using a calculator, we find that $\"root%283%2C3375%29=15\"$ . We could also notice that since 3*3*3*5*5*5=3375, we can just rearrange the terms to get $\"3%5E3%2A5%5E3=%283%2A5%29%5E3=15%5E3=3375\"$ which would mean that $\"15%5E3=3375\"$ (ie showing that 3375 is a perfect cube) \n" ); document.write( "

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