# Reduction from Partition problem to the same problem with the input replicated multiple times

In a variant of the Partition Problem given a multiset $$S = \{a_1,\dots,a_n\}$$ of positive integer with the total sum $$\sum_{i=1}^n a_i = m\,T$$, we want to find out if exists a partition $$S_1,\dots,S_m$$ of $$S$$ in which $$\sum_{a_i\in S_j} a_i = T$$ for each $$j$$.

Now my question is the following: given an instance $$S$$ of Partition, the instance that we construct by replicating the input $$k$$ times is equivalent to the initial instance?

To explain better from $$S$$ we construct a multiset $$S' = \bigcup_{i=1}^k S$$ in which the total sum is obviously $$\sum_{a_i \in S'} a_i = k\,m\,T$$ and in which we want to find out if exists a partiton $$S'_1,\dots,S'_{k\,m}$$ of $$S'$$ in which $$\sum_{a_i\in S'_j} a_i = T$$ for each $$j$$.

Is this new instance equivalent to the previous one? The case in which $$S$$ admits a partition is simple, because also $$S'$$ will admit a partition, but when $$S$$ does not admit a partition can we prove that also $$S'$$ does not admit one? or otherwise find a counterexample?