In essence, my problem boils down to finding all $i$ that satisfies this inequality ($n$ is constant):

$$
\frac{n}{i} \text{ (mod 1) } < \frac{n}{i+1} \text{ (mod 1) for }n,i\in\mathbb{N}, i < \sqrt{n}
$$

The problem that I face is that any sort of manipulation that I try to perform doesn't really make sense in the end, as cross-multiplying will result in $0$ on both sides as both sides will be integers and subtracting one side from the other will yield a useless expression in the end.

I also tried replacing the modulo with the subtraction of integers $p, q \in \mathbb{N}$ from each side of the inequality, but that lead me to a recursive solution for $i$.

Can anyone offer me any tips on how I can approach this problem? Any help is appreciated immensely!