In 1995, A, Wiles of the Princeton Institute for Advanced Study, announced, using cyclic groups ( a subject area which was not available at the time of Fermat), a proof of Fermat’s Last Theorem, which is stated as follows: If  is an odd prime and x, y,z are relatively prime positive integers, then = .    z  x  y In this paper, a proof of this theorem is given using only elementary Algebra. It is shown that if  is an odd prime and x, y,z are positive inyegera satisfying = ,    z x  y then x, y, and z are each divisible by .