The authors give three distinct proofs of the Gleissberg result that for \(r< m/2\), the number of partitions of \(n\) into parts congruent to \(\pm r \pmod m\) is equal to the number of partitions of \(n\) into parts congruent to \(0,\pm r\pmod m\), minimal difference \(m\) between parts, and no consecutive multiples of \(m\). One proof is combinatorial, one proof uses generating functions, and the third proof demonstrates that the numerator of the continued fraction \[ 1+ (a+b)q+ {abq^ 2 (1-q) \over 1+(a+b)q^ 2 + {abq^ 3 (1-q^ 2) \over 1+(a+b) q^ 3+ \cdots}} \] is \(\prod_ k^ \infty (1+aq^ k) (1+bq^ k)\).