- #1
Elwin.Martin
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Homework Statement
Let n be a fixed positive integer greater than 1. If a (mod n) = a' and b (mod n) = b', prove that (a+b) (mod n) = (a'+b') (mod n) and that (ab) (mod n) = (a'b') (mod n)
Homework Equations
When a = qn + r
a mod n = r
The Attempt at a Solution
(a'+b') (mod n) = (a (mod n) + b (mod n)) (mod n)
= ((a+b) (mod n)) (mod n)
= (a+b) (mod n)
(a'b') (mod n) = (a (mod n) * b (mod n)) (mod n)
= ((ab) mod n) mod n
= (ab) mod n
Is this a valid approach? My reasoning is that if we treat mod n as an operation on a number, then we can mod n twice and we should get the same thing since any remainder divided by the same number should yield the same number.
I know that my work isn't very rigorous and I didn't really apply the definition I have directly, can anyone point me in another direction if this is the wrong approach?
Thank you for your time,
Elwin