If 1 + i 1 - i m 2 = 1 + i i - 1 n 3 = 1 ,     m ,   n ∈ N then the greatest common divisor of the least values of m and n is

Step-by-step solution

1 + i 1 - i m 2 = 1 + i i - 1 n 3 = 1 Rationalize both the terms 1 + i 1 - i × 1 + i 1 + i m 2 = 1 + i i - 1 × - 1 - i - 1 - i n 3 = 1 1 + i 2 + 2 i 1 - i 2 m 2 = - 1 - i 2 - 2 i - 1 2 - i 2 n 3 = 1 We know i 2 = - 1 2 i 2 m 2 = - 2 i 2 n 3 = 1 i m 2 = - i n 3 = 1 For i m 2 = 1   ⇒ m 2 = 4 k   ⇒ m = 8 k ,   k ∈ I Since m is natural number so the least value of m is 8 . For - i n 3 = 1   ⇒ n 3 = 4 l   ⇒ n = 12 l ,   l ∈ I Since n is natural