A strong acid must have a weak conjugate base and a weak acid must have a strong conjugate base. The reason behind this is simple.
A "strong" acid or base is actually one that is unstable and will readily dissociate when added to water. A "weak" acid or base is one that is stable, and does not dissociate in water.
If an acid is to be "strong", it must readily dissociate to release hydrogen ions. However, the conjugate base that is formed must also be stable, so that is does not combine back with the released hydrogen ions, making it a weak base.