Q:

Let x, y be integers. What possible values can x^2 + y^2 take in Z4

Accepted Solution

A:
Answer:0, 1 or 2.Step-by-step explanation:An integer x in Z4 is either equal to [0], [1], [2] or [3] (as Z4 is made only of the remainders we can get when dividing an integer by 4).If x was equal to [0] in Z4, then x^2 = [0]*[0]=[0] in Z4.If x was equal to [1] in Z4, then x^2 = [1]*[1]=[1] in Z4.If x was equal to [2] in Z4, then x^2 = [2]*[2]=[4]=[0] in Z4 (as 4 and 0 are the same in Z4, given that both numbers leave a remainder of 0 when divided by 4).If x was equal to [3] in Z4, then x^2 = [3]*[3]=[9]=[1] in Z4 (as 9 and 1 are the same in Z4, given that both numbers leave a remainder of 1 when divided by 4).Therefore, in Z4 x^2+y^2 is either a sum of the form [0]+[0], or [0]+[1], or [1]+[0], or [1]+[1], which means we can only get either [0], [1] or [2].