document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210504>Question 1210504</A>: <I><SPAN STYLE=\"word-break: break-all;\"> If A and B are positive 2 digest integers, how many solutions are there of the equation 2a+3b=100 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853068 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13250)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><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=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=853068\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2a%2B3b=100 ALIGN=MIDDLE  ALT=\"2a%2B3b=100\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "This is a linear Diophantine equation -- one equation with two variables, with a finite number of solutions because the solutions are positive integers.  A common start on finding the set of solutions is to solve the equation for one variable.<br><BR>\n" );
document.write( "Because the numbers 2 and 100 are both even, it is probably fastest to solve the given equation for b.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3b=100-2a ALIGN=MIDDLE  ALT=\"3b=100-2a\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=%28100-2a%29%2F3 ALIGN=MIDDLE  ALT=\"b=%28100-2a%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=2%2850-a%29%2F3 ALIGN=MIDDLE  ALT=\"b=2%2850-a%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=2%28%2850-a%29%2F3%29 ALIGN=MIDDLE  ALT=\"b=2%28%2850-a%29%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > [1]<br><BR>\n" );
document.write( "With the equation in the last form [1], we see that (50-a) must be divisible by 3.  That means, among other things, that consecutive values of a that provide solutions will differ by 3.<br><BR>\n" );
document.write( "(1) Solution with smallest value of a...<br><BR>\n" );
document.write( "Remembering that the solutions must have both a and b positive 2-digit integers, we can see that the smallest 2-digit value for a that makes (50-a) divisible by 3 is 11.<br><BR>\n" );
document.write( "That gives us <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b=2%28%2850-11%29%2F3%29=2%2813%29=26 ALIGN=MIDDLE  ALT=\"b=2%28%2850-11%29%2F3%29=2%2813%29=26\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(2) Solution with the largest value of a...<br><BR>\n" );
document.write( "The solution with the largest value of a is the one with the smallest value of b.  Again since a and b must be 2-digit integers, equation [1] says that all values of b that give solutions will be even, so the smallest value of b that will give a solution is the smallest 2-digit even integer, which is 10.<br><BR>\n" );
document.write( "From (1) and (2), the solutions are the ones with b having even values from 10 to 26 inclusive.  The number of such solutions is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2826-10%29%2F2%2B1=9 ALIGN=MIDDLE  ALT=\"%2826-10%29%2F2%2B1=9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "ANSWER: There are 9 solutions having both a and b 2-digit positive integers<br><BR>\n" );
document.write( " </SPAN>\n" );
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