If a is any integer and a³ ≡ x (mod 9), then the possible values of x are:

Calculation :

Given a is any integer say a = 1,2,3,4,....

claim: Is to find the possible values of x if a3 ≡ x (mod 9).

So first .

if a = 1 then, 1³   1 (mod 9) ; in this case we get x = 1

if a = 2 then, 2³  8 (mod 9) ; in this case we get x = 8

if a = 3 then, 3³  9 (mod 9) 

                           0 (mod9) ; in this case we get x = 0

if a = 4 then, 4³  64 (mod 9) 

                           1 (mod 9) ; in this case we get x = 1

if a = 5 then, 5³  125 (mod 9)

                          8 (mod 9) ; in this case we get x = 8

if a = 6 then, 6³  216 (mod 9)

                          0 (mod 9) ; in this case we get x= 0 and so on ...

So, as we continue this process we get the same remainders as 0,1,8 repeatedly.

Hence, the possible values of x are 0, 1, 8.