Simple problem with no obvious solution strategy

[mathPuzzles]

We have some prime p and a given integer \(1\lt a \lt p-1\).

Some properties of simple multiples of a mod p are not straightforward to answer when they involve inequalities. Three independent examples of questions to think about (likely with related answers or strategies)

1) Given a constant \(N\lt p\), what is the smallest integer \(i>0\) which satisfies \(i a\mod p \lt N\)?

2) Over the range of integer i from 1 to N, what is the smallest value of \(i a \mod p\)?

3) For integer i in the range of \(1\lt i \lt N\), given another constant M, is \(i a \mod p\) ever less than M?


In all three cases you can solve them by replacing the inequality with equality, multiplying both sides by \(a^{-1} \mod p\) to find the corresponding value of i, then iterating over all possible values of N (or M). But surely there's a more efficient way to answer these questions, especially if N were impractically large for explicit enumeration.


Any suggestions how one might attack one or more of these problems? The inequality precludes the use of most modular algebra tricks.
Thanks!