Suppose $s$ and $t$ are coprime positive integers, and let $σ$ be an $s$-core partition and $τ$ a $t$-core partition. In this paper we consider the set $\mathcal P_{σ,τ}(n)$ of partitions of $n$ with $s$-core $σ$ and $t$-core $τ$. We find the smallest $n$ for which this set is non-empty, and show that for this value of $n$ the partitions in $\mathcal P_{σ,τ}(n)$ (which we call $(σ,τ)$-minimal partitions) are in bijection with a certain class of $(0,1)$-matrices with $s$ rows and $t$ columns. We then use these results in considering conjugate partitions: we determine exactly when the set $\mathcal P_{σ,τ}(n)$ consists of a conjugate pair of partitions, and when $\mathcal P_{σ,τ}(n)$ contains a unique self-conjugate partition.