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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

 

hence;

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

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

 

No problem so far.

 

Now;

 

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?

 

Secondly;

 

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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ac = bd + bkM + dhM + hkM^2

 

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

 

What is (bkM + dhM + hkM^2) mod M?

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0.........

 

So that makes ac = bd mod(M), then, right?

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