If is the imaginary cube root of unity, then find the number of pairs of integers (a, b) such that |a +b|=1

Step-by-step solution

We have |a +b|=1 aligned & |a +b|^2=1 (a +b)(a +b)=1 a^2+a b( + )+b^2=1 a^2-a b+b^2=1 & (a-b)^2+a b=1 ...(1) ( As 1+ + ^2=0) aligned when (a-b)^2=0 and a b=1 then (1,1) ;(-1,-1) when (a-b)^2=1 and a b=0 then (0,1) ;(1,0) ;(0,-1) ;(-1,0) Hence (0,1) ;(1,0) ;(0,-1) ;(-1,0) ;(1,1) ;(-1,-1) i.e. 6 ordered pairs