document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1148532>Question 1148532</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let n be a positive integer. If the equation 2x + 2y + z = n has 28 solutions in positive integers x, y and z, then what is the value of n? </SPAN>\n" );
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document.write( "Find the answers (there are two of them) by looking at the numbers of solutions for particular values of n.<br><BR>\n" );
document.write( "Suppose, for example, that n is 25.  Then we have<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2x%2B2y%2Bz+=+25 ALIGN=MIDDLE  ALT=\"2x%2B2y%2Bz+=+25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%28x%2By%29%2Bz+=+25 ALIGN=MIDDLE  ALT=\"2%28x%2By%29%2Bz+=+25\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "2(x+y) is even, and 25 is odd, so z must be odd.<br><BR>\n" );
document.write( "If z=1, then 2(x+y)=25-1=24, x+y=12, and there are 11 solutions in positive integers (x can be any integer from 1 to 11 inclusive).<br><BR>\n" );
document.write( "If z=3, then 2(x+y)=25-3=22, x+y=11, and there are 10 solutions in positive integers.<br><BR>\n" );
document.write( "If z=5, then 2(x+y)=25-5=20, x+y=10, and there are 9 solutions in positive integers.<br><BR>\n" );
document.write( "The pattern should be clear.  For n=25, the number of solutions in positive integers is 11+10+9+...+2+1 = 66.<br><BR>\n" );
document.write( "We want the number of solutions to be 28, which is 1+2+3+4+5+6+7.<br><BR>\n" );
document.write( "According to the pattern seen above, that will happen when z=1 and x+y=8; and that will be when n = 2(x+y)+z = 17.<br><BR>\n" );
document.write( "A similar investigation with n being even shows that the number of solutions will also be 28 when n is 18.<br><BR>\n" );
document.write( "ANSWERS: n is either 17 or 18 when the equation has 28 solutions.<br><BR>\n" );
document.write( " </SPAN>\n" );
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