# Let n be a fixed positive integer greater thatn 1 and let a and b be positive integers. Prove that a mod n = b mon n if and only if a = b mod.

Question
Matrix transformations
Let n be a fixed positive integer greater thatn 1 and let a and b be positive integers. Prove that a mod n = b mon n if and only if a = b mod.

2020-11-25
Since a mod n is a remaider when dividing a by n, we have that a mod n=a-kn
Similarly, b mod n=b-ln
Now notice that a mod n=b mod n if and only if a-kn=b-ln
which holds if and only if a-b=(k-l)n
Recall that, by definition, a=b mod n if and only if a-b=mn, where m is some integer.
Since k and l are integers, k-l is also an integer. Thus, a-b=(k-l)n
if and only if a=b mod n
Thus, we have proven that a mod n=b mod n if and only if a=b mod n, as required.

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