document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209235>Question 1209235</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the number of ordered pairs (m,n) of integers that satisfy<BR>\n" );
document.write( "mn = 3m + 3n + 17. </SPAN>\n" );
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document.write( "Answer: <font color=red>8</font>\r<BR>\n" );
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document.write( "Work Shown\r<BR>\n" );
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document.write( "mn = 3m+3n+17<BR>\n" );
document.write( "mn-3m-3n = 17<BR>\n" );
document.write( "mn-3m-3n+9 = 17+9<BR>\n" );
document.write( "(mn-3m)+(-3n+9) = 26<BR>\n" );
document.write( "m(n-3)-3(n-3) = 26<BR>\n" );
document.write( "(m-3)(n-3) = 26\r<BR>\n" );
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document.write( "n = 26 = 2^1*13^1 = p^a*q^b shows that a = 1, b = 1<BR>\n" );
document.write( "d(n) = number of positive divisors of n<BR>\n" );
document.write( "n = p^a*q^b<BR>\n" );
document.write( "d(n) = (1+a)*(1+b)<BR>\n" );
document.write( "d(26) = (1+1)*(1+1)<BR>\n" );
document.write( "d(26) = 4<BR>\n" );
document.write( "There are 4 positive divisors of 26 and they are: 1, 2, 13, 26.<BR>\n" );
document.write( "When including the negative divisors we get <font color=red>8</font> total divisors: <BR>\n" );
document.write( "1, 2, 13, 26, <BR>\n" );
document.write( "-1, -2, -13, -26\r<BR>\n" );
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document.write( "So there must be <font color=red>8 ordered pairs</font> (m,n) that make (m-3)(n-3) = 26 true and that make m*n = 3m+3n+17 true as well; where m,n are integers.<BR>\n" );
document.write( "If m,n were allowed to be decimal numbers, then there would be infinitely many ordered pair solutions. \r<BR>\n" );
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document.write( "Extra info:<BR>\n" );
document.write( "If you want to find each ordered pair, then set the factors m-3 and n-3 equal to the factors of 26.\r<BR>\n" );
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document.write( "Some examples:<BR>\n" );
document.write( "m-3 = 1 and n-3 = 26 lead to (m,n) = (4,29)<BR>\n" );
document.write( "m-3 = 2 and n-3 = 13 lead to (m,n) = (5,16)<BR>\n" );
document.write( "m-3 = 13 and n-3 = 2 lead to (m,n) = (16,5)\r<BR>\n" );
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document.write( "Here are all 8 integer solutions<BR>\n" );
document.write( "(-23,2), (-10,1), (1,-10), (2,-23), <BR>\n" );
document.write( "(4,29), (5,16), (16,5), (29,4)<BR>\n" );
document.write( "They are sorted from smallest x coordinate to the largest x coordinate.<BR>\n" );
document.write( "We have symmetry going on since (5,16) swaps to (16,5) as one example.\r<BR>\n" );
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document.write( "Side notes: <ul><li>x*y = 3x+3y+17 solves to y = (3x+17)/(x-3)</li><li>If you graphed y = (3x+17)/(x-3) using a tool like <a  rel=nofollow HREF=\"https://www.desmos.com/calculator\">Desmos</a> or <a  rel=nofollow HREF=\"https://www.geogebra.org/calculator\">GeoGebra</a> (among many others), you can then determine the locations of the lattice points. A lattice point (x,y) is where both x and y are integers.</li></ul><BR>\n" );
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