Can the speed up of the quadratic sieve algorithm be achieved for a divisor?

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The object is to finish this factorization

The last factor has $\ 123\ $ decimal digits , is composite and has almost surely no prime factor with less than $\ 40\ $ digits (I ran many ECM curves with high B1-values). It makes therefore sense to run the quadratic sieve.

It is well known that the quadratic sieve can be speed up dramatically, if the number has the form $\ a^b+c\ $ with "large" $\ b\ $ and "small" $\ c\ $.

Does this also hold for a divisor of such a number, or do I here just have to run the usual quadratic sieve ?