One day, the Tan family went out to a shopping centre where they bumped into Mrs Tan's secondary school friend. Mrs Tan's friend was surprised to see that Mrs Tan had five children. She asked, "How old are your children?" Mrs Tan's only daughter, Brenda, answered:
I am the middle child among the five of us.
We have five distinct positive integers for our ages which interestingly add up to 100.
Charles is my eldest brother. His age is a perfect square and a factor of 100.
Darius is immediately older than me while Alfred is immediately younger than me.
My youngest brother is Eric. My age is the geometric mean of Charles' and Eric's ages.
Exactly one of our ages is a prime number.
Can you help Mrs Tan's friend? Give your answer as the ages of the five children from the oldest to the youngest child.
You are encouraged to use a mathematical logic approach to solve this problem. Do not be swayed by what seems apparently logical or ridiculous.