document.write( "Question 1140911: The LCM of a and b is 12, and the LCM of b and c is 15. What is the value of the LCM
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Algebra.Com's Answer #761486 by ikleyn(52784) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "            The solution and the answer by @MathLover1 is INCORRECT.\r
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\n" ); document.write( "\n" ); document.write( "            I can easily give a counter-example.\r
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\n" ); document.write( "\n" ); document.write( "            Let a= 4, b= 3 and c= 5.\r
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\n" ); document.write( "\n" ); document.write( "            Then LCM(a,b) = LCM(4,3) = 12;\r
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\n" ); document.write( "\n" ); document.write( "                     LCM(b,c) = LCM(3,5) = 15 - so, the premise is satisfied;\r
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\n" ); document.write( "\n" ); document.write( "                     but LCM(a,c) = LCM(4,5) = 20, which contradicts to the solution by @Mathlover1.\r
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\n" ); document.write( "\n" ); document.write( "            I can easily give another counter-example.\r
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\n" ); document.write( "\n" ); document.write( "            Let a= 12, b= 3 and c= 5.\r
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\n" ); document.write( "\n" ); document.write( "            Then LCM(a,b) = LCM(12,3) = 12;\r
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\n" ); document.write( "\n" ); document.write( "                     LCM(b,c) = LCM(3,5) = 15 - so, the premise is satisfied;\r
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\n" ); document.write( "\n" ); document.write( "                     but LCM(a,c) = LCM(12,5) = 60, which contradicts to the solution by @Mathlover1.\r
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\n" ); document.write( "\n" ); document.write( "            The correct solution is below.\r
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document.write( "1.  The premise  LCM(a,b) = 12  and  LCM(b,c) = 15  implies that\r\n" );
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document.write( "        c is multiple of 5  and  b is not multiple of 4.\r\n" );
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document.write( "2.  Then EITHER \"a\" is 4  OR \"a\" is 12,\r\n" );
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document.write( "        and both/each of these two opportunities may have place.\r\n" );
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document.write( "3.  It implies that LCM(a,c) is EITHER  LCM(4,5)  = 20\r\n" );
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document.write( "                                OR      LCM(12,5) = 60,\r\n" );
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document.write( "    and, as my counter-examples above show, each and both these opportunities may have place.\r\n" );
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document.write( "4.  So, the answer to the problem's question is EITHER  20  OR  60.\r\n" );
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document.write( "    Each of these two opportunities may have place.\r\n" );
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