Answer: The three ages are 15, 17, 18.
Check: 15*17*18 = 4590 and all ages are distinct.
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Step-by-step explanation:
Let's say the three ages are x,y and z.
Since they are teenagers, this means x,y,z are values drawn from this set {13,14,15,16,17,18,19}
The number 4590 is even, so 2 is a factor. This means 2 is a factor of either x, y or z.
We see that 2 is a factor of 14,16 and 18. So it's possible that these are the x,y,z values we're after. But note how 5 is not a factor of any of these values, but it is a factor of 4590 (since the value ends in 0).
Furthermore, 3 is a factor of 4590 because the digits of 4590 add to a multiple of 3. We have 4+5+9+0 = 18 which is a multiple of 3.
The teenage years that are multiples of 3 are {15, 18}. Those two values either are even or have 3 as a factor or have 5 as a factor.
Let's see what we get when we multiply 15 and 18. So 15*18 = 270.
Divide 4590 over 270 to get 4590/270 = 17
Through luck we found the answer. Though usually with trial and error problems like this, it takes a bit more work.
So therefore the three ages are 15, 17, 18.
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Here's another approach:
Start with the set {13,14,15,16,17,18,19} which again is the set of all possible teen ages.
Divide 4590 over each item in that set
- 4590/13 = 353.077 approximately
- 4590/14 = 327.857 approximately
- 4590/15 = 306
- 4590/16 = 286.875
- 4590/17 = 270
- 4590/18 = 255
- 4590/19 = 241.579 approximately
We see that the denominators 15, 17, and 18 produce whole value results. Everything else leads to a non-whole decimal value. So we can eliminate 13, 14, 16, and 19.
The only thing left is 15,17,18 which are the three different ages.