The number 8 is the sum and product of the numbers in the collection of four positive integers $\{1,1,2,4\}$, since $1+1+2+4=8$ and $1 \times 1 \times 2 \times 4=8$. The number 2007 can be made up from a collection of $n$ positive integers that multiply to 2007 and add to 2007 . What is the smallest value of $n$ with $n>1$ ?
(A) 1171
(B) 1337
(C) 1551
(D) 1777
(E) 1781
To form such a collection of integers, our strategy is to include some integers larger than 1 whose product is 2007 and then add enough 1s to the collection to make the sum 2007.

In order to make $n$ as small as possible, we would like to include as few $1 \mathrm{~s}$ as possible, and so make the sum of the initial integers (whose product is 2007) as large as possible.

Since we would like to consider integers whose product is 2007 , we should find the divisors of 2007.

We see that $2007=3 \times 669=3 \times 3 \times 223$, and 223 is a prime number.

So the collections of positive integers larger than 1 whose product is 2007 are $\{3,669\},\{3,3,223\}$, and $\{9,223\}$.

The collection with the largest sum is $\{3,669\}$ whose sum is 672 . To get a sum of 2007 , we must add $2007-672=1335$ copies of 1 , which means that we are using $1335+2=1337$ integers. Therefore, the smallest value of $n$ is 1337 .

ANSWER: (B)