Minimise simultaneous equation remainder
This problem is an instance of linear programming.

If you allow fractional multiples then it can be solved in polynomial time using standard techniques for linear programming.

If you require integer multiples the problem is (strongly) $NP$-complete. There is a very easy reduction from 1-in-3-SAT. Let the dimension of the vectors be equal to the number of clauses, let the target vector be $(1,\ldots,1)$ and for every variabele create a vector $a_i$ that for each clause it appears in has a $1$ in that position (and a $0$ otherwise). You can achieve distance $0$ if and only if the formula is satisfiable.