PLHS - Maths

Let’s have two teasers on the go. 

What is the only number in which you can add up all it’s digits, and then multiply that number by three and get your original number?

Answer below, reason for your answer, and good luck once again!

X - 14 - 15 - 92 - 653 - 5897 - 9323 - Y - …..
What numbers should replace the X and the Y?
Answer in the comments box below, and again, a reason for your answer please!

Suppose you’re in a hallway with 100 closed lockers against the wall.
You start by opening every locker. Then you go back and close every second locker. Then you go to every third locker and open it if it’s closed, or close it if it’s open. Let’s call this action ‘toggling’ a locker. Continue ‘toggling’ every [...]

A trickier teaser for you all:
What are the next three numbers in this series? Again, a reason for your answer please!
4, 6, 12, 18, 30, 42, 60, 72, 102, 108

No, it’s not Mr Stewart.
Diophantus was a Greek mathematician who lived in the third century. He was one of the first mathematicians to use algebraic symbols.
Most of what is known about Diophantus’s life comes from an algebraic riddle from around the early sixth century. Stated simply, it says:
Diophantus’s youth lasted one sixth of his life. [...]