The product of three consecutive odd numbers is 9177 . What is the sum of the numbers?
(A) 51
(B) 57
(C) 60
(D) 63
(E) 69

## Part C: Each correct answer is worth 8.
Solution 1

Since $20 \times 20 \times 20=8000$ and $30 \times 30 \times 30=27000$, then we might guess that the three consecutive odd numbers whose product is 9177 are closer to 20 than they are to 30 .

Using trial and error, we determine that $21 \times 23 \times 25=12075$, which is too large.

The next smallest set of three consecutive odd numbers is 19,21,23 and the product of these three numbers is $19 \times 21 \times 23=9177$, as required.

Thus, the sum of the three consecutive odd numbers whose product is 9177 is $19+21+23=63$.

## Solution 2

We begin by determining the prime numbers whose product is 9177 .

(This is called the prime factorization of 9177.)

This prime factorization of 9177 is shown in the factor tree to the right. That is, $9177=3 \times 3059=3 \times 7 \times 437=3 \times 7 \times 19 \times 23$. Since $3 \times 7=21$, then $9177=21 \times 19 \times 23$ and so the three consecutive numbers whose product is 9177 are $19,21,23$.

Thus, the sum of the three consecutive odd numbers whose product is 9177 is $19+21+23=63$.

![](https://cdn.mathpix.com/cropped/2024_04_20_748fdafcde2a47412608g-190.jpg?height=358&width=315&top_left_y=401&top_left_x=1577)

ANSWER: (D)