ONLY NEED SOLUTION TO 5 B For any integer a prove the follow

ONLY NEED SOLUTION TO 5 B)

For any integer a, prove the following statements: (a) a^2 = 0 or 1 (mod 3). (b) a^3 = 0, 1 or -1 (mod 7). (c) a^4 = 0 or 1 (mod 5).

Solution

b) Let a be an integer.

Then

a=7q+r 0r<7

then,

a³=a*a*a=(7q+r )³=343q³+r³+21ar²+147a²r=(343q³+21ar²+147a²r)+r³

the terms inside the bracket are divisible by 7 ,so only r³ wil decise remainder

  0r³<7

the only integer value to satisfy this equation are r=0,-1,1

hence proved