SOLUTION: What is each of the five children's ages, if they are each born a year apart AND the product of their ages is 6,720?

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Question 772586: What is each of the five children's ages, if they are each born a year apart AND the product of their ages is 6,720?
Answer by KMST(5328) About Me  (Show Source):
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The ages will be either
A) 3 consecutive odd numbers along with the 2 even numbers in between,
or
B) 3 consecutive even numbers and the two odd numbers in between.

The prime factorization of 6,720 is
6720=2%5E6%2A3%2A5%2A7
We cannot regroup the prime factors into 5 factor as described in A) above, because alonng with 3, 5, and 7, we would need 6, and that would require an extra 3 as a prime factor.
WE must use 5 and 7 as our ofdd numbers and that means 4,6,and 8, as our even numbers.
6720=2%5E6%2A3%2A5%2A7=2%5E2%2A5%2A%282%2A3%29%2A7%2A2%5E3=highlight%284%2A5%2A6%2A7%2A8%29 is the product needed.
The children's ages are 4, 5, 6, 7, and 8.

We could have said
x= age of the middle child, so
x-2= age of the youngest child,
x-1= age of the second youngest child,
x%2B1 and x%2B2 being the ages of the two oldest ones.
Then the product would be
x%28x-1%29%28x%2B1%29%28x-2%29%28x%2B2%29=x%28x%5E2-1%29%28x%5E2-4%29
We could write x%28x%5E2-1%29%28x%5E2-4%29=6720 as an equation, but that is not very helpful.
At most we could have said
6720=x%28x%5E2-1%29%28x%5E2-4%29%3Cx%5E5,
and knowing that 5%5E5=3125%3C6720 would have suggested x%3E=6.