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Comments on Closed-form expression for sum of Modulo Arithmetic Progression

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Closed-form expression for sum of Modulo Arithmetic Progression

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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.

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1 comment thread

Do you intend the mod to be outside the sum or inside it—i.e., are you expecting the final result to ... (2 comments)
Do you intend the mod to be outside the sum or inside it—i.e., are you expecting the final result to ...
r~~‭ wrote over 1 year ago

Do you intend the mod to be outside the sum or inside it—i.e., are you expecting the final result to necessarily be less than M?

trichoplax‭ wrote over 1 year ago

It would be useful to have this answered by adding parentheses to the formula so it is immediately clear.