QJEEM20Ma71ef: JEE Main 2020 Mathematics - Class 11 JEEM20Ma71ef Solution: If ( 1 + i 1 − i ) m 2 = ( 1 + i 1 − i ) n 3 = 1 , (m, n ∈ N)…

Question

Verified Solution • 2025–26 Session moderate

If

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

, (m, n

\in

N) then the greatest common divisor of the least values of m and n is _______ .

Accepted Answer

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

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

\Rightarrow (i)^{m / 2} = (- i)^{n / 3} = 1

= i⁴ [ As i⁴ = 1]

\Rightarrow \frac{m}{2} = 4 k_{1} a n d \frac{n}{3} = 4 k_{2}

\Rightarrow

m = 8 k₁ and n = 12 k₂

Least value of m = 8 and n = 12.

∴

GCD (8, 12) = 4

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