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I was wondering if someone can explain to me a step in a proof given to me by my professor in regards to a modular operation theorem.

Addition theorem: Given three integers x, y, d (d > 0), (x+y)%d = (x%d + y%d) %d

Proof:

Let x = q(1)d + r(1) and y = q(2)d + r(2).

We have (x+y)%d = (q(1)d + r(1) + q(2)d + r(2)) %d

= (r(1) + r(2)) %d

Therefore: (x+y)%d = (x%d + y%d) %d

I dont get how my professor jumped from (q(1)d + r(1) + q(2)d + r(2))%d to (r(1) + r(2))%d.

Is there a specific reason for why we just ignore the product of q(1)d and q(2)d?

Thank you in advance.

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# Modular operation: Addition

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