Matrix approximation methods have successfully produced efficient, low-complexity approximate transforms for the discrete cosine transforms and the discrete Fourier transforms. For the DFT case, literature archives approximations operating at small power-of-two blocklenghts, such as \{8, 16, 32\}, or at large blocklengths, such as 1024, which are obtained by means of the Cooley-Tukey-based approximation relying on the small-blocklength approximate transforms. Cooley-Tukey-based approximations inherit the intermediate multiplications by twiddled factors which are usually not approximated; otherwise the effected error propagation would prevent the overall good performance of the approximation. In this context, the prime factor algorithm can furnish the necessary framework for deriving fully multiplierless DFT approximations. We introduced an approximation method based on small prime-sized DFT approximations which entirely eliminates intermediate multiplication steps and prevents internal error propagation. To demonstrate the proposed method, we design a fully multiplierless 1023-point DFT approximation based on 3-, 11- and 31-point DFT approximations. The performance evaluation according to popular metrics showed that the proposed approximations not only presented a significantly lower arithmetic complexity but also resulted in smaller approximation error measurements when compared to competing methods.