JEE PYQ: Complex Numbers Question 16
Question 16 - 2020 (03 Sep Shift 1)
If $\left(\frac{1+i}{1-i}\right)^{m/2} = \left(\frac{1+i}{i-1}\right)^{n/3} = 1$, $(m, n \in \mathbb{N})$, then the greatest common divisor of the least values of $m$ and $n$ is ______.
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Answer: 4
Solution
$\frac{1+i}{1-i} = i$, so $i^{m/2} = 1 \Rightarrow m/2 = 4k$, minimum $m = 8$. $\frac{1+i}{i-1} = \frac{(1+i)^2}{-2} = -i$, so $(-i)^{n/3} = 1 \Rightarrow n/3 = 4k$, minimum $n = 12$. $\gcd(8, 12) = 4$.
BETA
