Question 1192792
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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>
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>
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>
Let's look at just one letter to
(1) find the numerical representation of the letter;
(2) find the encryption (encrypted letter) for that number; and
(3) convert that encrypted number back to a letter<br>
The first letter in the message is M.<br>
(1) the way I read your post, M is 12, not 13.<br>
(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>
Those two parts are straightforward.<br>
For the decryption, we need to find the number n for which 7n+3 is equal to 9 plus some multiple of 26:<br>
{{{7n+3=26k+9}}}
{{{7n=26k+6}}}
{{{7n=21k+5k+6}}}
{{{n=3k+(5k+6)/7}}}<br>
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>
6, 11, 16, 21<br>
So<br>
5k+6=21
5k=15
k=3
26k+9=78+9=87
7n+3=87
7n=84
n=12<br>