document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192792>Question 1192792</A>: <I><SPAN STYLE=\"word-break: break-all;\"> MAT 145: Topics In Contemporary Math\r<BR>\n" );
document.write( "\n" );
document.write( "           QUESTION 12\r<BR>\n" );
document.write( "\n" );
document.write( "Convert the message “MAC IS A CAT” using the encryption e = (7m + 3) mod 26.  A table showing number values for the letters is below.\r<BR>\n" );
document.write( "\n" );
document.write( "A\r<BR>\n" );
document.write( "\n" );
document.write( "B\r<BR>\n" );
document.write( "\n" );
document.write( "C\r<BR>\n" );
document.write( "\n" );
document.write( "D\r<BR>\n" );
document.write( "\n" );
document.write( "E\r<BR>\n" );
document.write( "\n" );
document.write( "F\r<BR>\n" );
document.write( "\n" );
document.write( "G\r<BR>\n" );
document.write( "\n" );
document.write( "H\r<BR>\n" );
document.write( "\n" );
document.write( "I\r<BR>\n" );
document.write( "\n" );
document.write( "J\r<BR>\n" );
document.write( "\n" );
document.write( "K\r<BR>\n" );
document.write( "\n" );
document.write( "L\r<BR>\n" );
document.write( "\n" );
document.write( "M\r<BR>\n" );
document.write( "\n" );
document.write( "0\r<BR>\n" );
document.write( "\n" );
document.write( "1\r<BR>\n" );
document.write( "\n" );
document.write( "2\r<BR>\n" );
document.write( "\n" );
document.write( "3\r<BR>\n" );
document.write( "\n" );
document.write( "4\r<BR>\n" );
document.write( "\n" );
document.write( "5\r<BR>\n" );
document.write( "\n" );
document.write( "6\r<BR>\n" );
document.write( "\n" );
document.write( "7\r<BR>\n" );
document.write( "\n" );
document.write( "8\r<BR>\n" );
document.write( "\n" );
document.write( "9\r<BR>\n" );
document.write( "\n" );
document.write( "10\r<BR>\n" );
document.write( "\n" );
document.write( "11\r<BR>\n" );
document.write( "\n" );
document.write( "12\r<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "N\r<BR>\n" );
document.write( "\n" );
document.write( "O\r<BR>\n" );
document.write( "\n" );
document.write( "P\r<BR>\n" );
document.write( "\n" );
document.write( "Q\r<BR>\n" );
document.write( "\n" );
document.write( "R\r<BR>\n" );
document.write( "\n" );
document.write( "S\r<BR>\n" );
document.write( "\n" );
document.write( "T\r<BR>\n" );
document.write( "\n" );
document.write( "U\r<BR>\n" );
document.write( "\n" );
document.write( "V\r<BR>\n" );
document.write( "\n" );
document.write( "W\r<BR>\n" );
document.write( "\n" );
document.write( "X\r<BR>\n" );
document.write( "\n" );
document.write( "Y\r<BR>\n" );
document.write( "\n" );
document.write( "Z\r<BR>\n" );
document.write( "\n" );
document.write( "13\r<BR>\n" );
document.write( "\n" );
document.write( "14\r<BR>\n" );
document.write( "\n" );
document.write( "15\r<BR>\n" );
document.write( "\n" );
document.write( "16\r<BR>\n" );
document.write( "\n" );
document.write( "17\r<BR>\n" );
document.write( "\n" );
document.write( "18\r<BR>\n" );
document.write( "\n" );
document.write( "19\r<BR>\n" );
document.write( "\n" );
document.write( "20\r<BR>\n" );
document.write( "\n" );
document.write( "21\r<BR>\n" );
document.write( "\n" );
document.write( "22\r<BR>\n" );
document.write( "\n" );
document.write( "23\r<BR>\n" );
document.write( "\n" );
document.write( "24\r<BR>\n" );
document.write( "\n" );
document.write( "25\r<BR>\n" );
document.write( "\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "Find the numerical representation of the message.                                        <BR>\n" );
document.write( " \r<BR>\n" );
document.write( "\n" );
document.write( "Find the encryption for the numbers.                                                               <BR>\n" );
document.write( "  \r<BR>\n" );
document.write( "\n" );
document.write( "Convert the encrypted numbers back to letters.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #824827 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13198)</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=824827\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "In this message, although the presentation is formatted very poorly, it appears that you are defining A=0, B=1, ..., and Z=25.  That would be in line with the fact that you are doing an encryption mod 26, with the results being from 0 to 25 instead of from 1 to 26.<br><BR>\n" );
document.write( "And that would make my responses to the other questions you submitted incorrect -- but that was because you didn't define your questions clearly.<br><BR>\n" );
document.write( "The process here is straightforward; there is no point in our doing the whole problem for you.  But there is one part of your question that you might need some help with.<br><BR>\n" );
document.write( "Let's look at just one letter to<BR>\n" );
document.write( "(1) find the numerical representation of the letter;<BR>\n" );
document.write( "(2) find the encryption (encrypted letter) for that number; and<BR>\n" );
document.write( "(3) convert that encrypted number back to a letter<br><BR>\n" );
document.write( "The first letter in the message is M.<br><BR>\n" );
document.write( "(1) the way I read your post, M is 12, not 13.<br><BR>\n" );
document.write( "(2) the encryption of 12 is (7*12+3) mod 26 = (87) mod 26 = 9.  That would apparently correspond to the 10th letter in the alphabet, which is J.<br><BR>\n" );
document.write( "Those two parts are straightforward.<br><BR>\n" );
document.write( "For the decryption, we need to find the number n for which 7n+3 is equal to 9 plus some multiple of 26:<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7n%2B3=26k%2B9 ALIGN=MIDDLE  ALT=\"7n%2B3=26k%2B9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7n=26k%2B6 ALIGN=MIDDLE  ALT=\"7n=26k%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=7n=21k%2B5k%2B6 ALIGN=MIDDLE  ALT=\"7n=21k%2B5k%2B6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n=3k%2B%285k%2B6%29%2F7 ALIGN=MIDDLE  ALT=\"n=3k%2B%285k%2B6%29%2F7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "n and 3k are integers, so (5k+6)/7 has to be an integer.  The elementary (and fastest) way to find k is to list the integers of the form 5k+6 to find the smallest one that is a multiple of 7:<br><BR>\n" );
document.write( "6, 11, 16, 21<br><BR>\n" );
document.write( "So<br><BR>\n" );
document.write( "5k+6=21<BR>\n" );
document.write( "5k=15<BR>\n" );
document.write( "k=3<BR>\n" );
document.write( "26k+9=78+9=87<BR>\n" );
document.write( "7n+3=87<BR>\n" );
document.write( "7n=84<BR>\n" );
document.write( "n=12<br><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );