# Affine transformations and letter frequencies

Number Theory Level pending

Let $$A \rightarrow 0, B \rightarrow 1, \ldots , Z \rightarrow 25$$. 

The most common letters in the English alphabet are $$E$$ and $$T$$.
The most common letters in a long ciphertext , enciphered by an affine transformation $$a * P + b \equiv C \pmod{26}$$ are $$M$$ and $$X$$, respectively.
We guess that $$M$$ and $$X$$ correspond to the two most common letters in the English alphabet $$E$$ and $$T$$.

$$a * P + b \equiv C \bmod{26} \implies P \equiv \bar{a} * (C - b)$$, where $$\bar{a}$$ is the inverse of $$a$$ modulo 26.

Find $$( \bar{a} + b) \bmod{26}$$.

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