[thaumasiotes]

> d is a common divisor of a,b if there exists x,y such that dx = ay

    d = 9
    a = 3
    b = 537
    x = 1
    y = 3

    dx = 9(1) = 9
    ay = 3(3) = 9 = dx

You can't actually have meant this? You're claiming that 9 is a common divisor of the pair (3, b), where b is any value. It's not even a divisor of 3.