Question Number 104333 by 175mohamed last updated on 20/Jul/20

if i^( 5n +1 ) = 1    , n∈Z  then show that : n = 3+4p   ,p∈Z

Answered by OlafThorendsen last updated on 20/Jul/20

i^(5n+1)  = 1 = i^(4k)   ⇔ 5n+1 = 4k  ⇔ n = −1+4k−4n  ⇔ n = 3+4(k−n−1)  ⇔ n = 3+4p

Answered by mr W last updated on 21/Jul/20

i=cos (π/2)+i sin (π/2)  i^(5n+1) =cos (((5n+1)π)/2)+i sin (((5n+1)π)/2)  1=cos 2kπ+i sin 2kπ  ⇒(((5n+1)π)/2)=2kπ  ⇒5n+1=4k  ⇒n=4k−4n−1  ⇒n=4(k−n−1)+3  ⇒n=4p+3