N=(a+ib)^3−107i
N=a^3−3ab^2 +i(3a^2b−b^3-107)
Since N is a positive integer
Im(N)=0
⇒3a^2b−b^3=107
⇒3a^2−b^2= 107/b
​
3a^2b^2 is an integer [∵a,b are integers]
⇒b divider 107
case (1) : b=107
⇒a^2= [(107)^2+1]/3
​ ⇒a^2= (11+50)/3
​
Not possible because a cannot be integers here

case (2) b=1
⇒3a^2=108
⇒a=6
⇒N=6^3−3×6×1
⇒N=198

There are two positive integers A and B, and we need to find the value of n, given that n equals A^3B - 107i, where i is a positive integer.

Calculating n:

To find the value of n, we need to determine the values of A and B. Let's break down the equation step by step.

1. A^3B - 107i
2. A^3B is a positive integer, so let's assume A^3B = x, where x is a positive integer.
3. Therefore, the equation becomes x - 107i.

Finding the value of n:

To find the value of n, we need to determine the value of x.

1. We know that n = x - 107i.
2. Since n is a positive integer, x - 107i must also be a positive integer.
3. In other words, x must be greater than 107i.

Understanding the equation:

The equation A^3B - 107i = x implies that x is a positive integer greater than 107i. Let's analyze this equation further.

1. A^3B represents a cube of a positive integer multiplied by another positive integer B.
2. By multiplying A^3 by B, we obtain a positive integer.
3. Subtracting 107i ensures that the result remains a positive integer.

Conclusion:

In conclusion, the value of n is determined by the positive integers A and B, such that n equals A^3B - 107i, where i is a positive integer. To find the value of n, we need to determine the values of A and B, ensuring that A^3B is a positive integer greater than 107i.