Monique Henson

2023-04-01

The product of the ages, in years, of three (3) teenagers os 4590. None of the have the sane age. What are the ages of the teenagers???

meakQueueksiu7e

To solve this problem, we need to find three different numbers whose product is 4590. We can start by factoring the number 4590 to find its prime factors. Once we have the prime factors, we can determine how to distribute them among the three teenagers' ages.
The prime factorization of 4590 is:
$4590=2·{3}^{3}·5·17$
Now, let's distribute these prime factors among the three teenagers' ages. We want to make sure that each teenager has a different age.
We can start with the largest prime factor, which is 17. We assign 17 to one of the teenagers, let's call their age $x$.
Next, we assign the remaining prime factors to the other two teenagers. We have $2·{3}^{3}·5$ left to distribute. We can split this among the two remaining teenagers. Let's call their ages $y$ and $z$.
Since we want the product of their ages to be 4590, we can set up the following equation:
$x·y·z=4590$
Substituting the prime factorization, we have:
$17·y·z=4590$
Now, we need to find two numbers whose product is 4590 when multiplied by 17. Let's try different combinations until we find a solution.
We can start by trying $y=2$ and $z={3}^{3}·5$.
$17·2·\left({3}^{3}·5\right)=17·2·{3}^{3}·5=4590$
Therefore, we have found a solution where one teenager is 17 years old, another teenager is 2 years old, and the third teenager is ${3}^{3}·5=135$ years old.
So, the ages of the three teenagers are 17, 2, and 135, which multiply together to give 4590.
In conclusion, the ages of the teenagers are 15, 17, and 18 because 4590 is divisible by 15, 17, and 18.

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