Question

If a and b are positive integer such that N = (a + ib)³ – 10⁷ i is a positive integer. Find N.

Answer 1

N = (a + ib)³ – 107i

= a³ – ib³ + 3a² ib – 3ab² – 107i

N = (–b³ + 3a²b – 107)I + a³ – 3ab²

⇒3a²b – b³ = 107

⇒b(3a² – b²) = 107

⇒b =13a² – b² = 107

⇒a² = 36

⇒N = a³ – 3ab² = 216 – 18 = 198