document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195807>Question 1195807</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The number 35 has the property that when its digits are both increased by 2, and then multiplied, the result is 5 times 7=35, equal to the original number.<BR>\n" );
document.write( "Find the sum of all two-digit number such that when you increase both digits by 2, and then multiply these numbers, the product is equal to the original number. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #828353 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</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=828353\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "Let x be the tens digit of the 2-digit number and y be the units digit.<br><BR>\n" );
document.write( "Then the number itself is 10x+y.<br><BR>\n" );
document.write( "Adding 2 to each digit and multiplying the two resulting numbers gives the expression (x+2)(y+2).<br><BR>\n" );
document.write( "The conditions of the problem are that<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B2%29%28y%2B2%29=10x%2By ALIGN=MIDDLE  ALT=\"%28x%2B2%29%28y%2B2%29=10x%2By\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "subject to the constraints that x and y are single-digit integers.<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%2B2%29%28y%2B2%29=10x%2By ALIGN=MIDDLE  ALT=\"%28x%2B2%29%28y%2B2%29=10x%2By\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy%2B2x%2B2y%2B4=10x%2By ALIGN=MIDDLE  ALT=\"xy%2B2x%2B2y%2B4=10x%2By\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Gather all the terms in one variable on one side of the equation and all terms without that variable on the other, then simplify:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy%2B2x-10x=y-2y-4 ALIGN=MIDDLE  ALT=\"xy%2B2x-10x=y-2y-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=xy-8x=-y-4 ALIGN=MIDDLE  ALT=\"xy-8x=-y-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Factor out the variable on the left and divide:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%28y-8%29=-y-4 ALIGN=MIDDLE  ALT=\"x%28y-8%29=-y-4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-y-4%29%2F%28y-8%29 ALIGN=MIDDLE  ALT=\"x=%28-y-4%29%2F%28y-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Perform the division as quotient plus remainder.  To do this, given a \"-y\" term in the numerator and \"y-8\" as the denominator, rewrite the numerator as a multiple of (y-8) plus a constant; then perform the division:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28%28-y%2B8%29-12%29%2F%28y-8%29 ALIGN=MIDDLE  ALT=\"x=%28%28-y%2B8%29-12%29%2F%28y-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=%28-y%2B8%29%2F%28y-8%29-12%2F%28y-8%29 ALIGN=MIDDLE  ALT=\"x=%28-y%2B8%29%2F%28y-8%29-12%2F%28y-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=-1-12%2F%28y-8%29 ALIGN=MIDDLE  ALT=\"x=-1-12%2F%28y-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "In this form of the equation, -1 is an integer and x has to be an integer; that means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12%2F%28y-8%29 ALIGN=MIDDLE  ALT=\"-12%2F%28y-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be an integer.  Note also that, because x must be a positive integer, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-12%2F%28y-8%29 ALIGN=MIDDLE  ALT=\"-12%2F%28y-8%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be a positive integer greater than 1.  That means y-8 must be negative and a factor of 12.<br><BR>\n" );
document.write( "Try (y-8) equal to each negative number that is a factor of 12 to find solutions:<br><BR>\n" );
document.write( "(1) y-8 = -12 --> y = -4  no; y must be positive<BR>\n" );
document.write( "(2) y-8 = -6  --> y = 2; x = -1 + (-12/-6) = 1.  The number is 12; (1+2)(2+2) = 3*4 = 12  YES<BR>\n" );
document.write( "(3) y-8 = -4 --> y = 4; x = -1 + (-12/-4) = 2.  The number is 24; (2+2)(4+2) = 4*6 = 24  YES<BR>\n" );
document.write( "(4) y-8 = -3 --> y = 5; x = -1 + (-12/-3) = 3.  The number is 35; (3+2)(5+2) = 5*7 = 35  YES<BR>\n" );
document.write( "(5) y-8 = -2 --> y = 6; x = -1 + (-12/-2) = 5.  The number is 56; (5+2)(6+2) = 7*8 = 56 YES<BR>\n" );
document.write( "(6) y-8 = -1 --> y = 7; x = -1 + (-12/-1) = 10  no; x must be a single digit<br><BR>\n" );
document.write( "We have four numbers that satisfy the conditions of the problem: 12, 24, 35, and 56.<br><BR>\n" );
document.write( "ANSWER: 12+24+35+56 = 127<br>\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );