Question

The product of the ages of three teenagers is 2730, and their ages are integers. The sum of their ages

(A) is less than 40.
(B) is 41.
(C) is 42.
(D) is 51.
(E) cannot be determined.

Answer 1

Explanation:

We can start by prime factorizing 2730 as $2730 = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 13$. Since the three ages are integers, we can pair up the prime factors in three different ways to get three possible sets of ages:

Age Set 1: $2 \cdot 3 \cdot 5 = 30$, $7$, and $13$. The sum of these ages is $30 + 7 + 13 = 50$.

Age Set 2: $2 \cdot 3 \cdot 7 = 42$, $5$, and $13$. The sum of these ages is $42 + 5 + 13 = 60$.

Age Set 3: $2 \cdot 5 \cdot 7 = 70$, $3$, and $13$. The sum of these ages is $70 + 3 + 13 = 86$.

Therefore, the sum of their ages could be 50, 60, or 86. None of the answer choices match any of these values exactly, but we can see that the sum of their ages is closest to 51, which is answer choice (D). So our final answer is (D) is the closest choice to the sum of the ages of the three teenagers.