In the first Harry Potter book, it is said that Flamel and Dumbledore invented the philosopher's stone together. It's also heavily implied that this stone is the only reason why Flamel lived so long. This means that Dumbledore is… let's say less than 200 years younger than Flamel, to be safe.
But since Flamel's age is given and is somewhere in the 600s, that would mean that Dumbledore is more than 400 years old. But after the stone was destroyed, he did not die from that and it's also not said that he had a stash of the elixir.
In book 5 there's even a short appearance of someone who was an examinant while Dumbledore was still in school, meaning she would have to be older than him. I would be very surprised if she also got the elixir of life.
The same with Dumbledore's brother: Did he get the elixir? Or their parents and he's multiple centuries younger?
Or are there other ways that wizards become many centuries old? In that case I would wonder why an elixir of life was even necessary.
The only explanations I can think of are these:
- Dumbledore wasn't actually involved in creating the stone and book 1 lied.
- Flamel isn't actually that old and book 1 lied (and maybe others as well).
- Rowling is very bad at Math, much more than usual.