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If the plaintext is made of both letters (a to z) and digits (0 to 9), how do you find the key domain of the multiplication cipher?

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You have 36 possible "characters" here.

If a single character is encrypted by E(C) = (c * k) % 36 then possible keys k are numbers that are coprime to 36, ie. gcd(k,36)=1. Furthermore it makes not much sense to consider numbers not between 1 and 36, because of the modulo.

As 36=2*2*3*3, the possible keys are basically all numbers not multiples of 2 and/or 3. That is:
5 7 11 13 17 19 23 25 29 31 35
Technically 1 too, but this would be no change from plaintext

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