An analysis of partitions is made by considering the congruence of sets of numbers, which is to some extent equivalent to, but more far-reaching than Nakayama's method of hooks. A proof is given of Nakayama's conjecture concerning the p-blocks of symmetric group characters which is much simpler than the recent proof by Brauer and Robinson. The p-residue and p-quotient of a partition are defined, and a formula is obtained relating to symmetric group characteristics. A procedure is described whereby the mode of separation in every case may be determined, of the 0-characters of the symmetric groups into p-characters. The method of p-residues and p-quotients is employed to give a method of expansion for the plethysm of S-functions which is unambiguous and yet reasonably concise.