The least positive integer n such that $\frac{(2i)^{n}}{(1-i)^{n-2}}$, $i=\sqrt{-1}$, is a positive integer, is ____.

JEE Main · 2021 · 26 Aug 2021 · Shift 2 · Complex Numbers

The least positive integer n such that

\frac{( 2 i ) ^{n}}{( 1 - i ) ^{n - 2}}

i = - 1

, is a positive integer, is ____.

Answer: 6

With

2 i = 2 e^{iπ /2}

and

1 - i = 2 e^{- iπ /4}

\frac{( 2 i ) ^{n}}{( 1 - i ) ^{n - 2}} = \frac{2 ^{n} e ^{inπ /2}}{2 ^{(n - 2) /2} e ^{- i (n - 2) π /4}} = 2^{(n + 2) /2} e^{i (3 n - 2) π /4}

Its modulus

2^{(n + 2) /2}

is positive for every

n

, so the expression is a positive real number exactly when

\frac{( 3 n - 2 ) π}{4}

is a multiple of

2 π

, that is when

3 n - 2 \equiv 0 (mod 8)

. Multiplying by

3

modulo

8

gives

n \equiv 6 (mod 8)

, and the smallest positive solution is

n = 6

, for which the value is

2^{4} = 16

. The least positive integer

n

6

JEE Main · 2021 · 26 Aug 2021 · Shift 2 · Integer answer