There are $126$ boxes containing apples. The number of apples in each box lies between $120$ and $150$ (inclusive). Let $F(n)$ indicate the number of boxes containing $n$ apples and $m$ indicates the maximum value of $F(n)$ for any $n$. What could be the minimum possible value of $m$?
Pigeons$=126$
Pigeonholes$=31$
$m=126/31=5$
Maximum value=$5$.
How to find minimum?
This is a gmat exam question.
If every value would be less than $5$, the sum would be at most $4\cdot 31=124$. Hence your $m=5$ is already the solution.
Note that $29$ times value $4$ and $2$ times value $5$ realizes the $126$ boxes.