document.write( "Question 541373: The positive integers 30, 72, and N have the property that the product of any two of them is divisible by the third. What is the smallest possible value of N? \n" ); document.write( "

Algebra.Com's Answer #354181 by Edwin McCravy(20056) $\"\"$ $\"About$

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document.write( "30, 72, and N\r\n" );
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document.write( "There must exist positive integers A, B, and C such that \r\n" );
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document.write( "30·72 = NA\r\n" );
document.write( "  30N = 72B\r\n" );
document.write( "  72N = 30C\r\n" );
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document.write( "Simplifying:\r\n" );
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document.write( "2160 =  NA\r\n" );
document.write( "  5N = 12B\r\n" );
document.write( " 12N =  5C  \r\n" );
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document.write( "Solving each for N:\r\n" );
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document.write( "N = \r\n" );
document.write( "N = \r\n" );
document.write( "N = \r\n" );
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document.write( "Set the last two right sides equal, since both are equal to N\r\n" );
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document.write( " = \r\n" );
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document.write( "144B = 25C\r\n" );
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document.write( "The smallest B and C can be are B = 25 and C = 144\r\n" );
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document.write( "Substituting in\r\n" );
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document.write( "N =  =  = 60 \r\n" );
document.write( "N =  =  = 60\r\n" );
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document.write( "That works because\r\n" );
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document.write( "30, 72, and 60\r\n" );
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document.write( "30*72 = 2160 = 60*36\r\n" );
document.write( "30*60 = 1800 = 73*25\r\n" );
document.write( "60*72 = 4320 = 30*144\r\n" );
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document.write( "So the answer is N = 60\r\n" );
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document.write( "Edwin

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