If (1+i1-i)^m/2=(1+ii-1)^n/3=1, (n,m N), then the greatest common divisor of the least val…

Question

If $\left(\dfrac{1+i}{1-i}\right)^{m/2}=\left(\dfrac{1+i}{i-1}\right)^{n/3}=1$, ($n,m\in\mathbb N$), then the greatest common divisor of the least values of $m$ and $n$ is

Answer: 4

Accepted Answer

$\dfrac{1+i}{1-i}=i$, $\dfrac{1+i}{i-1}=-i$. So $i^{m/2}=(-i)^{n/3}=1$, giving least $m=8$, least $n=12$.

$$\gcd(8,12)=4.$$

Solutions · 1