document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199210>Question 1199210</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+a%5E2b%5E3c%5E4=648000+ ALIGN=MIDDLE  ALT=\"+a%5E2b%5E3c%5E4=648000+\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where a, b and c are distinct positive integers greater than 1, what is the least possible value of a+b+c? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #832967 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=832967\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Prime factorization:<BR>\n" );
document.write( "648000 = 2^6*3^4*5^3\r<BR>\n" );
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document.write( "Then rewrite the expression a bit.<BR>\n" );
document.write( "648000 = 2^6*3^4*5^3<BR>\n" );
document.write( "648000 = (2^3)^2*5^3*3^4<BR>\n" );
document.write( "648000 = 8^2*5^3*3^4<BR>\n" );
document.write( "It fits the format a^2*b^3*c^4 such that<BR>\n" );
document.write( "a = 8<BR>\n" );
document.write( "b = 5<BR>\n" );
document.write( "c = 3<BR>\n" );
document.write( "So a+b+c = 8+5+3 = 16\r<BR>\n" );
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document.write( "We can do a bit of rearranging like this<BR>\n" );
document.write( "648000 = 2^6*3^4*5^3<BR>\n" );
document.write( "648000 = 2^4*2^2*3^4*5^3<BR>\n" );
document.write( "648000 = 2^4*(2*3^2)^2*5^3<BR>\n" );
document.write( "648000 = 2^4*18^2*5^3<BR>\n" );
document.write( "648000 = 18^2*5^3*2^4<BR>\n" );
document.write( "Showing that<BR>\n" );
document.write( "a+b+c = 18+5+2 = 25<BR>\n" );
document.write( "which isn't smaller than the previous result (16) we computed.\r<BR>\n" );
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document.write( "Or we could do this<BR>\n" );
document.write( "648000 = 2^6*3^4*5^3<BR>\n" );
document.write( "648000 = 2^2*2^4*3^4*5^3<BR>\n" );
document.write( "648000 = 2^2*(2*3)^4*5^3<BR>\n" );
document.write( "648000 = 2^2*6^4*5^3<BR>\n" );
document.write( "648000 = 2^2*5^3*6^4<BR>\n" );
document.write( "So,<BR>\n" );
document.write( "a+b+c = 2+5+6 = 13<BR>\n" );
document.write( "which is smaller than the 16 computed earlier.\r<BR>\n" );
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document.write( "I used computer software to do a brute-force search of all possible triples a,b,c from 2 to 100. Those are the only three possible sums I could find. There may be others.\r<BR>\n" );
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document.write( "Answer: 13<BR>\n" );
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