document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=175629>Question 175629</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/0028253345/algebrahomeworkh>Algebra 1</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> If the digits of a two-digit positive interger are reversed, the result is 6 less than twice the original numder. Find all such intergers for which this is true.\r<BR>\n" );
document.write( "\n" );
document.write( "How do you figure this out? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #130725 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=130725\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let the tens digit be represented by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5Bt%5D ALIGN=MIDDLE  ALT=\"d%5Bt%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the ones digit be represented by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5Bo%5D ALIGN=MIDDLE  ALT=\"d%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  That means you can represent any two digit positive number by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10d%5Bt%5D+%2B+d%5Bo%5D ALIGN=MIDDLE  ALT=\"10d%5Bt%5D+%2B+d%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > if you restrict <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5Bt%5D ALIGN=MIDDLE  ALT=\"d%5Bt%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5Bo%5D ALIGN=MIDDLE  ALT=\"d%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > to the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If our original number is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10d%5Bt%5D+%2B+d%5Bo%5D ALIGN=MIDDLE  ALT=\"10d%5Bt%5D+%2B+d%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > then the number with reversed digits is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10d%5Bo%5D+%2B+d%5Bt%5D ALIGN=MIDDLE  ALT=\"10d%5Bo%5D+%2B+d%5Bt%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Then the conditions of the problem give us:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2810d%5Bt%5D+%2B+d%5Bo%5D%29+-+6+=+10d%5Bo%5D+%2B+d%5Bt%5D ALIGN=MIDDLE  ALT=\"2%2810d%5Bt%5D+%2B+d%5Bo%5D%29+-+6+=+10d%5Bo%5D+%2B+d%5Bt%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Simplify and solve for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5Bt%5D ALIGN=MIDDLE  ALT=\"d%5Bt%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=20d%5Bt%5D+%2B+2d%5Bo%5D+-+6+=+10d%5Bo%5D+%2B+d%5Bt%5D ALIGN=MIDDLE  ALT=\"20d%5Bt%5D+%2B+2d%5Bo%5D+-+6+=+10d%5Bo%5D+%2B+d%5Bt%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=19d%5Bt%5D+=+8d%5Bo%5D+%2B+6 ALIGN=MIDDLE  ALT=\"19d%5Bt%5D+=+8d%5Bo%5D+%2B+6\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5Bt%5D+=+%288d%5Bo%5D+%2B+6%29%2F19 ALIGN=MIDDLE  ALT=\"d%5Bt%5D+=+%288d%5Bo%5D+%2B+6%29%2F19\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now remember that both <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5Bt%5D ALIGN=MIDDLE  ALT=\"d%5Bt%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5Bo%5D ALIGN=MIDDLE  ALT=\"d%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > are restricted to the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.  That means that we have to find which of the given set of numbers, when substituted for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5Bo%5D ALIGN=MIDDLE  ALT=\"d%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >, yield <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5Bt%5D ALIGN=MIDDLE  ALT=\"d%5Bt%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > also an element of the given set.  We know that since the denominator is 19 and there are only ten consecutive integer possibilities, we have at most 1 solution.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "We can also see that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8d%5Bo%5D+%2B+6 ALIGN=MIDDLE  ALT=\"8d%5Bo%5D+%2B+6\"   DISABLED_style= \"max-width:90%; height:auto;\" > must be an integer multiple of 19.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The first five integer multiples of 19 are 19, 38, 57, 76, and 95.  So:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8d%5Bo%5D+%2B+6+=+19 ALIGN=MIDDLE  ALT=\"8d%5Bo%5D+%2B+6+=+19\"   DISABLED_style= \"max-width:90%; height:auto;\" > &#8594; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8d%5Bo%5D+=+12 ALIGN=MIDDLE  ALT=\"8d%5Bo%5D+=+12\"   DISABLED_style= \"max-width:90%; height:auto;\" > &#8594; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5Bo%5D ALIGN=MIDDLE  ALT=\"d%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > not an integer.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8d%5Bo%5D+%2B+6+=+38 ALIGN=MIDDLE  ALT=\"8d%5Bo%5D+%2B+6+=+38\"   DISABLED_style= \"max-width:90%; height:auto;\" > &#8594; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8d%5Bo%5D+=+32 ALIGN=MIDDLE  ALT=\"8d%5Bo%5D+=+32\"   DISABLED_style= \"max-width:90%; height:auto;\" > &#8594; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=red%28d%5Bo%5D+=+4%29 ALIGN=MIDDLE  ALT=\"red%28d%5Bo%5D+=+4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > IS an integer.  This is our solution, but let's continue just to make sure.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8d%5Bo%5D+%2B+6+=+57 ALIGN=MIDDLE  ALT=\"8d%5Bo%5D+%2B+6+=+57\"   DISABLED_style= \"max-width:90%; height:auto;\" > &#8594; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8d%5Bo%5D+=+51 ALIGN=MIDDLE  ALT=\"8d%5Bo%5D+=+51\"   DISABLED_style= \"max-width:90%; height:auto;\" > &#8594; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5Bo%5D ALIGN=MIDDLE  ALT=\"d%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > not an integer.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8d%5Bo%5D+%2B+6+=+76 ALIGN=MIDDLE  ALT=\"8d%5Bo%5D+%2B+6+=+76\"   DISABLED_style= \"max-width:90%; height:auto;\" > &#8594; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8d%5Bo%5D+=+70 ALIGN=MIDDLE  ALT=\"8d%5Bo%5D+=+70\"   DISABLED_style= \"max-width:90%; height:auto;\" > &#8594; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5Bo%5D ALIGN=MIDDLE  ALT=\"d%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > not an integer.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8d%5Bo%5D+%2B+6+=+95 ALIGN=MIDDLE  ALT=\"8d%5Bo%5D+%2B+6+=+95\"   DISABLED_style= \"max-width:90%; height:auto;\" > &#8594; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8d%5Bo%5D+=+89 ALIGN=MIDDLE  ALT=\"8d%5Bo%5D+=+89\"   DISABLED_style= \"max-width:90%; height:auto;\" > &#8594; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5Bo%5D ALIGN=MIDDLE  ALT=\"d%5Bo%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > not an integer AND <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5Bo%5D+%3E+9 ALIGN=MIDDLE  ALT=\"d%5Bo%5D+%3E+9\"   DISABLED_style= \"max-width:90%; height:auto;\" > so we can stop looking.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now we know the ones digit is 4, so using <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%5Bt%5D+=+%288%284%29+%2B+6%29%2F19+=+38%2F19+=+2 ALIGN=MIDDLE  ALT=\"d%5Bt%5D+=+%288%284%29+%2B+6%29%2F19+=+38%2F19+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" > we know the tens digit is 2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The only positive two-digit integer that satisfies the given conditions is 24.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Check the answer:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "2 times 24 is 48 which is 6 more than 42 which is 24 with the digits reversed.  Checks.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "By the way, the word is integer, not inte<big><b>r</big></b>ger.\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );