A supermarket has $128$ crates of apples. Each crate contains at least $120$ apples and at most $144$ apples. 
What is the largest integer $n$ such that there must be at least $n$ crates containing the same number of apples?
$\textbf{(A) }4\qquad \textbf{(B) }5\qquad \textbf{(C) }6\qquad \textbf{(D) }24\qquad  \textbf{(E) }25$
To find the largest number of "repeated" crates necessary, we must account for all the possibilities of the number of apples in each crate. Since each crate contains a minimum of $120$ apples and a maximum of $144$ apples, there are $144 - 120 + 1 = 25$ different amounts possible for the number of apples per crate.
Now, we have to count for the worst case scenario: the $25$ amounts are repeated as many times as possible.
$25$ can go into $128$ exactly $5$ times because $5 \cdot 25 = 125$, which is less than $128$. This leaves a remainder of $3$ crates.
The worst case scenario would be that these $3$ crates have a different number of apples each. It doesn't actually matter how many apples are in these $3$ crates because any of the $25$ values would be repeated again anyway. So, the answer is $5 + 1 = \boxed{\textbf{(C) }6}$ ~[jiang147369](https://artofproblemsolving.comhttps://artofproblemsolving.com/wiki/index.php/User:Jiang147369)