If a and b are positive integers such that N = a + i b 3 - 107 i (where N is a natural number), then the value of a is equal to (where i 2 = - 1 )

Step-by-step solution

a + i b 3 - 107 i = Natural number ⇒ a 3 - 3 b 2 a + i 3 a 2 b - b 3 - 107 i = Natural number ⇒ 3 a 2 b - b 3 - 107 = 0 ⇒ a 2 = b 3 + 107 3 b = b 2 3 + 107 3 b ⇒ b 3 + 107 is a multiple of 3 and b is a factor of 107 ⇒ b = 1 and a 2 = 108 3 = 36 ⇒ a = 6 b cannot be equal to 107 , because 107 3 + 107 is not a multiple of 3