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 [...]

All of the gold coins are identical, and of the same value.
If I divide the coins into two unequal numbers (or piles), then 32 times the difference between the two numbers equals the difference between the squares of the two numbers.
How many coins do I have? Remember, a reason for your answer please!

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