Writing It Up · Challenge problem

A student claims: "if $a \equiv b \pmod n$ then $a^2 \equiv b^2 \pmod{n^2}$", with the proof "$a = b + kn$, so $a^2 = b^2 + 2bkn + k^2n^2$." Where the audit asks "would this also prove something false?", what is the smallest $n \ge 2$ for which the claim fails?