Question

If a is any integer and a^3 = x (mod 9), then the possible values of x are:​

Answer 1

Step-by-step explanation:

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