Different positive integers can be written in the eight empty circles so that the product of any three integers in a straight line is 3240 . What is the largest possible sum of the eight numbers surrounding 45 ?
(A) 139
(B) 211
(C) 156
(D) 159
(E) 160

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-088.jpg?height=405&width=411&top_left_y=966&top_left_x=1288)

## Part C: Each correct answer is worth 8.
Since the three numbers in each straight line must have a product of 3240 and must include 45, then the other two numbers in each line must have a product of $\frac{3240}{45}=72$.

The possible pairs of positive integers are 1 and 72, 2 and 36,3 and 24,4 and 18, 6 and 12, and 8 and 9 .

The sums of the numbers in these pairs are 73, 38, 27, 22, 18, and 17 .

To maximize the sum of the eight numbers, we want to choose the pairs with the largest possible sums, so we choose the first four pairs.

Thus, the largest possible sum of the eight numbers is $73+38+27+22=160$.

ANswer: (E)