Am I misunderstanding the problem? It is clear that we want at most $1$ bag per box (since there are a fixed number of objects in each box). Let us use $a$ red boxes and $b$ blue boxes. We have $43a+17b=731$. We wish to minimize $a+b$. We have $17n=43(17-a)$. We must thus have $17\mid {a}$. So either $a=0$ or $a=17$ The minimum number of bags is then just $17$.

277546 wrote: Am I misunderstanding the problem? I think the question meant that the bags have to be able to be sorted into 17 boxes of 43 and 43 boxes of 17 - using 17 bags of 43 wouldn't work since you wouldn't be able to split the bags into 43 bags of 17.