For each $k\ge 1$ there is a sequence $x_{1,k},\ldots,x_{k,k}$ of positive integers such that $$ \sum_{i=1}^k a_i x_{i,k} =2\sum_{i=1}^k x_{i,k} \quad\Longrightarrow\quad a_1=\cdots= a_k=2 $$ (For $k=3$ the minimal example seems to be $(9, 12, 17)$.)

This is easy to prove by induction using only knowledge of modular arithmetic. But is there a more general theory that this result fits into?

For each $k\ge 1$ there is a sequence $x_{1,k},\ldots,x_{k,k}$ of positive integers such that for all nonnegative integers $ a_i $, $$ \sum_{i=1}^k a_i x_{i,k} =2\sum_{i=1}^k x_{i,k} \quad\Longrightarrow\quad a_1=\cdots= a_k=2 $$ (For $k=3$ the minimal example seems to be $(9, 12, 17)$.)

This is easy to prove by induction using only knowledge of modular arithmetic. But is there a more general theory that this result fits into?