1976 AMC 12 Problem 12

Count the possible apple totals from $120$ through $144$, inclusive

If every total occurred at most five times, there would be at most $25\cdot5$ crates

There are $144-120+1=25$ possible apple counts. Since $128>25\cdot5,$ some count occurs at least $6$ times. This is sharp: distribute $128$ crates so that three counts occur $6$ times and the other twenty-two occur $5$ times. Thus the largest guaranteed $n$ is $6.$

Therefore, the correct answer is C.