The sum of four different positive integers is 100 . The largest of these four integers is $n$. The smallest possible value of $n$ is
(A) 26
(B) 50
(C) 28
(D) 27
(E) 94

## Part B: Each correct answer is worth 6.
Suppose that the integers $a25$.

Since $n$ is an integer, then $n$ is at least 26 .

Could $n$ be 26? In this case, we would have $a+b+c=100-26=74$.

If $n=26$, then $a+b+c$ is at most $23+24+25=72$, which means that we cannot have $a+b+c=74$.

Therefore, $n$ cannot be 26 .

Could $n$ be 27? In this case, we would have $a+b+c=100-27=73$.

Here, we could have $a+b+c=23+24+26=73$, and so $n=27$ is possible, which means that the smallest possible value of $n$ is 27 . (There are other values of $a, b, c$ that work with $n=27$ as well.)

ANSWER: (D)