Here's a very simple way to emulate a do-while loop:

condition = True
while condition:
    # loop body here
    condition = test_loop_condition()
# end of loop

The key features of a do-while loop are that the loop body always executes at least once, and that the condition is evaluated at the bottom of the loop body. The control structure show here accomplishes both of these with no need for exceptions or break statements. It does introduce one extra Boolean variable.

The program prints at least one factor of an integer y, or if it has no factors (other than itself and 1), prints that y is prime.

It uses the variable x to try all possible factors greater than one. It starts at the floor of y divided by 2, because no number larger than half of y could be a factor. Using normal division rather than floor division could give you a fractional value if y is odd. (An even better solution is to start with the square root of y - if y is not prime, one of its factors will be less than or equal to its square root.)

Inside the loop, it tests y % x, which is the remainder after dividing y by x. If the remainder is zero, that means that x is a factor of y, and it prints it.

The else clause is executed at the end of the loop, unless a factor is found, in which case the "break" skips out of the loop and the else clause. So either a factor is found, or it's prime.

Here's the improved code with the indentation fixed:

import math

def check_primality(y):
  x = int(math.sqrt(y))
  while x > 1:
    if y % x == 0:                                                
      print y, 'has factor', x
      break
    x -= 1
  else:
    print y, 'is prime'