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Modular Arithmetic


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I'm struggling to understand what is a proof in modular arithmetic and what does not. Example below;


a = b(mod M) c=d(mod M)


1) Prove: a + c = b + d (mod M);


a = b + hM

c = d + kM



a + c = b + d (h +k)M

a + c = b + d (mod M).


No problem so far.




1) ac = bd (mod M)


ac = (b + hM)(d + kM)

ac = bd + bkM + dhM + hkM^2


To me- that isnt ac = bd (mod M) -> what am I missing?




a - c = b - d + (h-k)M


which to me IS a correct modular arithemtic calculation i.e. a-c=b-d (mod M).

Aparantly it's not thought?

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