An elementary and transparent representation of the fast Fourier transform is given. Instead of using the usual and highly algebraic approach it is shown how a Fourier transform of the ordern=p·m can be reduced top Fourier transforms of orderm by performing essentiallym Fourier transforms of orderp on the data. The resulting process is discussed in more detail forn=3q andn=5q. The problem of retrieval of the wanted coefficients from the final data is solved by a simple argument. The generalization for an ordern equal to a product of powers of prime numbers is immediate.