What you found was actually the "order of 11". Euler's theorem says that is always AN exponent that makes the statement true. But not necessarily the smallest (the order of the integer). Whenever the order is that is called a "primitive roots" (in an algebraic way of sense we can think of this a generator of the cyclic group). The number 45 has not primitive roots, as a problem solved by Gauss. It is not one of the standard forms for which primitive roots exists. (primes, power of primes, and doubling power of primes).