document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=499201>Question 499201</A>: <I><SPAN STYLE=\"word-break: break-all;\"> determine the smallest 6 digit number which is exactly divisible by 8,15,and21<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #337465 by </B><A HREF=/tutors/aboutme.mpl?userid=oberobic><B>oberobic(2304)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=oberobic><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=337465\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> What are the factors of 8?<BR>\n" );
document.write( "1, 2, 4<BR>\n" );
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document.write( "What are the factors of 15?<BR>\n" );
document.write( "1, 3, 5<BR>\n" );
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document.write( "What are the factors of 21?<BR>\n" );
document.write( "1, 3, 7<BR>\n" );
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document.write( "The domain set is:  {1, 2, 4, 3, 5, 7}<BR>\n" );
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document.write( "What is the least common multiple that is equal to or greater than 21 (the largest number)?<BR>\n" );
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document.write( "1*2*4*3*5*7 = 840<BR>\n" );
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document.write( "840/8 = 105<BR>\n" );
document.write( "840/15 = 56<BR>\n" );
document.write( "840/21 = 40<BR>\n" );
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document.write( "So, 840 is looking good!<BR>\n" );
document.write( "We can check 420, but we'll find it does not work for 8.<BR>\n" );
document.write( "420/8 is not an integer solution.<BR>\n" );
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document.write( "But what does the question ask?  The smallest <b>6-digit number</b> that is exactly divisible by 8, 15, and 21.<BR>\n" );
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document.write( "How about 840,000?<BR>\n" );
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document.write( "840,000/8 = 105,000<BR>\n" );
document.write( "840,000/15 = 56,000<BR>\n" );
document.write( "840,000/21 = 40,000<BR>\n" );
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document.write( "That works, but is it the <b>smallest</b> 6-digit number?<BR>\n" );
document.write( "Not likely.<BR>\n" );
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document.write( "Looking back, 8*15*21*x, where x is an integer, will always be divisible exactly by 8, 15 & 21.<BR>\n" );
document.write( "We want to find an x such that the product is > 100,000 by the smallest margin.<BR>\n" );
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document.write( "8*15*21=2520<BR>\n" );
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document.write( "2520x >= 100,000<BR>\n" );
document.write( "x >= 39.6824397<BR>\n" );
document.write( "The next integer is 40.<BR>\n" );
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document.write( "40*2520 = 100,800<BR>\n" );
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document.write( "100,800/8 = 12,600<BR>\n" );
document.write( "100,800/15 = 6,720<BR>\n" );
document.write( "100,800/21 = 4,800<BR>\n" );
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document.write( "The smallest 6-digit number (that is, a number >= 100,000) that is exactly divisible by 8, 15, and 21 is 100,800.<BR>\n" );
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document.write( "Done.<BR>\n" );
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