What is the smallest positive integer n for which

$(1+i)^{2 n}=(1-i)^{2 n}$

$\Rightarrow\left[(1+i)^{2}\right]^{n}=\left[(1-i)^{2}\right]^{n}$

$\Rightarrow\left(1^{2}+i^{2}+2 i\right)^{n}=\left(1^{2}+i^{2}-2 i\right)^{n}$

$\Rightarrow(1-1+2 i)^{n}=(1-1-2 i)^{n} \quad\left[\because i^{2}=-1\right]$

$\Rightarrow(2 i)^{n}=(-2 i)^{n}$

$\Rightarrow(2 i)^{n}=(-1)^{n}(2 i)^{n}$

$\Rightarrow(-1)^{n}=1$

$\Rightarrow n$ is a multiple of 2

Thus, the smallest positive integer $n$ for which $(1+i)^{2 n}=(1-i)^{2 n}$ is 2 .