Describe the following groups and find the quadratic residue

Describe the following groups and find the quadratic residues in each of them. U_23. U_24. U_33.

Solution

Definition of quadratic residue .Let p be an integer , if there is an integer x such that 0< x < p

and the equation x² = q (mod p) has a solution then q iscall ed the quadratic residue of \' p \'

U₂₃ is the set of all quadratic residue of the integer 23

1. U ₂₃ = { 1, 2, 3, 4, 6, 8 ,9, 12, 13 ,16,18} if we put x= 1,2,3...... we get the remainders as 1,2,3,4,6,8,9,12,13,16,18, and the remainders will not have the values 5,7 ,10 ,11 , 14 ,15, 17, 19, 20 21, 22

2. U ₂₄= { 1,4,9,12,16}

3 . U_{33 = { 1,3,4,9,12,15,16,22,25,27,31}}