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Closed-form expression for sum of Modulo Arithmetic Progression

~~Is there any closed-form expression or atleast an efficient way to calculate this sum?~~
$$ \sum_{i=1}^{N} (a \cdot i) \bmod{M} $$
we can assume $N$, $a$, and $M$ are large enough such that simple looping is not feasible and that period of the progression is also large.
I am aware of the way to calculate sum of first $N$ modulo natural numbers, but I am not able to extend that idea to this series.

Is there any closed-form expression or at least an efficient way to calculate this sum?
$$ \sum_{i=1}^{N} (a \cdot i) \bmod{M} $$
we can assume $N$, $a$, and $M$ are large enough such that simple looping is not feasible and that period of the progression is also large.
I am aware of the way to calculate sum of first $N$ modulo natural numbers, but I am not able to extend that idea to this series.

formula summation sequence modular-arithmetic

formula summation sequence modular-arithmetic