Show that if [a] in Z/mZ is a zero divisor, it cannot have an inverse in Z/mZ.?

For a zero divisor a, there exists b ≠ 0 such that ab = 0.

Now suppose a has an inverse a⁻¹. Then

b = (1)b = (a⁻¹a)b = a⁻¹(ab) = a⁻¹(0) = 0,

contradicting the above assumption on b.

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