Groups: Work on Problems D9-D11 with a partner. Problem D11 is quite challenging, and you may approach it any number of ways, either by using "data" from their secret words, or by trying to solve the equation for P.
Remember that there is no "divide by 3" rule in a mod 26 system. If you come up with the equation P = (C - 2) / 3 (mod 26), how would you decipher M? M = 12, so P = (12 - 2) / 3 = 10 / 3. How can you find 10 / 3 in this system?
Groups: Before wrapping up this part of the session, share your equations and solutions. If no one actually solved the equation for P, you can look at this solution on an overhead or on the board:
C = 3P + 2 (mod 26)
C - 2 (mod 26) = 3P (mod 26)
Notice that we need mod 26 on both sides at this point, because although C is between 0 and 25, C - 2 may not be (it may be -2, for example).
Now you need to multiply both sides of the equation by the reciprocal of 3. There is no 1/3 in a mod 26 system, but there is a number r, so that 3r = 1 (mod 26). Discuss what that number must be (the answer is 9, because 3 * 9 = 27 = 1 (mod 26)).
9 * (C - 2) (mod 26) = 9 * (3P) (mod 26)
P = 9 * (C - 2) (mod 26)
If you worked on this another way, you may have come up with different but equivalent equations:
P = 9 C - 18 (mod 26)
P = 9 C +12 (mod 26)
and so on.
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