Why didn't this work:
(11,26) is 1
11 is prime so phi of 11 is 10 but 11 to the 10 isn't congruent to 1 mod 26. It is 11 to the 12.
Correct me if I'm wrong, but what you tried to do was:
$\displaystyle \gcd(11,26)=1,\text{ so }11^{\phi(11)}\equiv1\mod26.$
This is not correct.
Euler's Theorem states:
$\displaystyle \text{ If }\gcd(a,m)=1,\text{ then }a^{\phi(m)}\equiv1\mod m$.
See?