please help

Miss Minerva challenges her class to guess her secret positive integer. She gives them these clues:
1) The sum of the digits of this three-digit number is 14.
2) The tens digit is the cube of a prime number.
3) The units digit is one-fifth of the hundreds digit.
What is Miss Minerva's number?

Hello class from Miss Minerva!

$x+y+z=14\\ y\in\{{\mathbb{P}^3}\}\\ z=\frac{x}{5}$

$y\in\{{\mathbb{P}^3}\}=14-x-z\\ y\in\{{\mathbb{P}^3}\}=14-x-\frac{x}{5}\\ y\in\{{\mathbb{P}^3}\}=14-5-1$

$x=5\\ y=8\\ z=1$

  !