# Residues distribution modulo an interval

Given a number $$n$$ and an Interval $$I = [ \; \lfloor n^{1/4} \rfloor, \lfloor n^{(1/3) \rfloor \;} ]$$, can we say anything about the distribution of $$\{ n \mod b \;\;| \; b \in I \}$$?

1. In particular, if I wanted the residue to be close to any region in $$[0, b-1]$$, say close to the "top" of the residue classes around $$b-1$$ could I choose a $$b$$ for which that is the case without doing much work? By not doing much work I mean not having to try every single $$b$$ in the interval $$I$$ until I find one.

2. Could I somehow choose better $$b$$'s in my search than just randomly trying them; or even better, just choose the right $$b$$ in one go.

The factorization of $$n$$ is known if that helps any.

Heuristically, it seems most residues tend to be close to the "top", but I just need to $$b$$'s where I will know where the residues will land without having to try all of them.

Note that $$n \equiv b-1 \pmod b$$ implies $$b | n+1$$, $$n \equiv b-2 \pmod b$$ implies $$b | n+2$$, and so on. Therefore you can factor $$n+1$$, $$n+2$$, and so on, until one of them has a factor in $$I$$. You can do something similar for other target regions.
While I don't have a proof, empirically it seems like you only need to factor $$O(1)$$ numbers on average, because around half of the numbers $$n$$ I've checked have a divisor in the range $$[n^{\frac14},n^{\frac13}]$$.