Let a, b be in [a]m. So, b = a + mk for some integer k.
Then, for β to be well-defined, we need [a]n = [b]n = [a + mk]n.
==> [a + mk]n - [a]n = [0]n
==> [mk]n = [0]n
==> We need m to be a multiple of n (since k can be an arbitrary integer).
I hope this helps!
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Verified answer
Let a, b be in [a]m. So, b = a + mk for some integer k.
Then, for β to be well-defined, we need [a]n = [b]n = [a + mk]n.
==> [a + mk]n - [a]n = [0]n
==> [mk]n = [0]n
==> We need m to be a multiple of n (since k can be an arbitrary integer).
I hope this helps!