JCSBimp's Hexagonal Orthogonal

It now seems that the task at hand is this: Given two linear equations with integer coefficients:

y₁ = a₁x + b₁, and y₂ = a₂x + b₂,

find the integer values of x, if any, where y₁ is an integer multiple of y_2.

y₁ = n*y₂ when x = (b₂n – b₁)/(a₁ – a₂n).

Currently, lacking insight, it seems finding n is no easier than finding x. But I am not yet giving up. For m = pq where p and q are distinct odd primes, p < q, and m = s² – r where s is the smallest integer greater than the square root of m, factorization of m will reduce to finding integer values of x producing values of pairs of linear equations as above, where a₁ = s, a₂ = -1, b₁ = -r, and b₂ = s. If we consider p = s – c₁ and q = s + c₂ for positive integers c₁ and c₂, finding positive integer x such that y₂ divides y₁ will give us c₁ = x and c₂ = y₁/y₂ and, through this, the factorization of m.