document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=499191>Question 499191</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The sum of four natural numbers a1, a2, a3 and a4 is \r<BR>\n" );
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document.write( "210. What is the minimum possible LCM of these 4 mentioned \r<BR>\n" );
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document.write( "natural numbers?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #337479 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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/cgi-bin/show-face.mpl?user=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=337479\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Note that 210 = 2*3*5*7. Without loss of generality assume that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a_1 \le a_2 \le a_3 \le a_4\">. The optimal case would be \"a1, a2, a3 are factors of a4\" in which the LCM would be simply a4. If this is true, then we can construct casework in which we find the minimum possible value of a4.\r<BR>\n" );
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document.write( "If a4 is strictly greater than a3, then a3 is at most 1/2 of a4, and\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a_1 + a_2 + a_3 + a_4 \le \frac{1}{2}a_4 + \frac{1}{2}a_4 + \frac{1}{2}a_4 + a_4 = \frac{5}{2}a_4 \Rightarrow \frac{5}{2}a_4 \ge 210 \Rightarrow a_4 \ge 84\">. The optimal case would be 84, 42, 42, 42.\r<BR>\n" );
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document.write( "The next case would be \"a4 = a3\" and a2 is strictly less. Then,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a_1 + a_2 + a_3 + a_4 \le \frac{1}{2}a_4 + \frac{1}{2}a_4 + a_4 + a_4 = 3a_4 \Rightarrow 3a_4 \ge 210 \Rightarrow a_4 \ge 70\">\r<BR>\n" );
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document.write( "Or, if a4 = a3 = a2, then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE a_1 + a_2 + a_3 + a_4 \le \frac{1}{2}a_4 + a_4 + a_4 + a_4 = \frac{7}{2}a_4 \Rightarrow \frac{7}{2}a_4 \ge 210 \Rightarrow a_4 \ge 60\">\r<BR>\n" );
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document.write( "We can check that {30, 60, 60, 60} satisfies, and their LCM is 60. </SPAN>\n" );
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