If $\left(\frac{1+\mathrm{i}}{1-\mathrm{i}}\right)^{\mathrm{m} / 2}=\left(\frac{1+\mathrm{i}}{\mathrm{i}-1}\right)^{\mathrm{n} / 3}=1, \quad(\mathrm{m}, \mathrm{n} \in \mathrm{N})$ Then the greatest common divisor of the least values of $m$ and $n$ is ____.

JEE Main · 2020 · 03 Sep 2020 · Shift 1 · Complex Numbers

(\frac{1 + i}{1 - i})^{m /2} = (\frac{1 + i}{i - 1})^{n /3} = 1, (m, n \in N)

Then the greatest common divisor of the least values of

m

and

n

is ____.

Answer: 4

Since

\frac{1 + i}{1 - i} = i

and

\frac{1 + i}{i - 1} = - i

, the conditions read

i^{m /2} = 1

and

(- i)^{n /3} = 1

. As

i^{4} = 1

, the first needs

\frac{m}{2}

to be a multiple of

4

, so the least

m

8

; as

(- i)^{4} = 1

, the second needs

\frac{n}{3}

to be a multiple of

4

, so the least

n

12

. The greatest common divisor of

8

and

12

4

JEE Main · 2020 · 03 Sep 2020 · Shift 1 · Integer answer