document.write( "Question 1129602: The product of all the positive divisors of a positive integer N is 6*16*36*96. What is N? \n" ); document.write( "
Algebra.Com's Answer #746204 by Edwin McCravy(20056)\"\" \"About 
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The product of all the positive divisors of a positive integer N is 6∙16∙36∙96. What is N?
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document.write( "We prime factor those factors:\r\n" );
document.write( "6 = 2∙3\r\n" );
document.write( "16 = 2∙2∙2∙2\r\n" );
document.write( "36 = 2∙2∙3∙3\r\n" );
document.write( "96 = 2∙2∙2∙2∙2∙3\r\n" );
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document.write( "Therefore 6∙16∙36∙96 = 21234 \r\n" );
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document.write( "From above we see that N contains prime factors 2 and 3 and ONLY those two\r\n" );
document.write( "prime factors. Now we must determine how many times N contains each as a\r\n" );
document.write( "factor. \r\n" );
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document.write( "N cannot contain the factor 3 more than once. Because:  If so, N would\r\n" );
document.write( "contain 2∙3∙3 as a factor.  Then the factors of N would contain factors \r\n" );
document.write( "1, 2, 3, 2∙3, 3∙3, 2∙3∙3, which would produce a product of factors 23∙36.\r\n" );
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document.write( "Then the product of factors of N would contain 3 as a factor 6 times.\r\n" );
document.write( "However we know that the product of factors of N contains 3 as a factor only\r\n" );
document.write( "4 times. \r\n" );
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document.write( "Therefore N contains 3 as a factor exactly once, no more and no less!\r\n" );
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document.write( "So we only need determine how many times N contains 2 as a factor.\r\n" );
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document.write( "It must contain 2 as a factor more than once, for if N contained 2 as a\r\n" );
document.write( "factor only once, then N would be 2∙3=6, with product of factors\r\n" );
document.write( "(1)(2)(3)(2∙3) = 22∙33, not 21234.\r\n" );
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document.write( "N also must contain 2 as a factor more than twice, for if it contained 2 as\r\n" );
document.write( "a factor only twice, then N would be 2∙2∙3=12, with product of factors\r\n" );
document.write( "(1)(2)(3)(2∙2)(2∙3)(2∙2∙3) = 26∙33, not 21234.\r\n" );
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document.write( "So we try N as containing 2 as a factor 3 times.  That would make N = 2∙2∙3\r\n" );
document.write( "= 24. Then the product of factors of N would be\r\n" );
document.write( "(2)(3)(2∙2)(2∙3)(2∙2∙2)(2∙2∙3)(2∙2∙2∙3) = 21234, which\r\n" );
document.write( "is exactly what we're looking for!\r\n" );
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document.write( "So the answer is N = 24.\r\n" );
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document.write( "Edwin
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