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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