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

y_{1} = a_{1}x + b_{1}, and y_{2} = a_{2}x + b_{2},

find the integer values of x, if any, where y_{1} is an integer multiple of y_{2.}

y_{1} = n*y_{2} when x = (b_{2}n – b_{1})/(a_{1} – a_{2}n).

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 = s^{2} – 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 a_{1} = s, a_{2} = -1, b_{1} = -r, and b_{2} = s. If we consider p = s – c_{1} and q = s + c_{2} for positive integers c_{1} and c_{2}, finding positive integer x such that y_{2} divides y_{1} will give us c_{1} = x and c_{2} = y_{1}/y_{2} and, through this, the factorization of m.

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