Find all the square roots of x^2 = 53 (mod 77) by hand. 2 marks

Accepted Solution

Answer:[tex]x=\pm\sqrt{77n+53}[/tex]Step-by-step explanation:Given : [tex]x^2\equiv 53\mod 77[/tex]To find : All the square roots ?Solution :The primitive roots modulo is defined as[tex]a\equiv b\mod c[/tex]Where, a is reminderb is dividend c is divisor  Converting equivalent into equal,[tex]a-b=nc[/tex]Applying in [tex]x^2\equiv 53\mod 77[/tex],[tex]x^2\equiv 53\mod 77[/tex][tex]x^2-53=77n[/tex][tex]x^2=77n+53[/tex][tex]x=\pm\sqrt{77n+53}[/tex]We have to find the possible value in which the x appear to be integer.The possible value of n is 4.As [tex]x=\pm\sqrt{77(4)+53}[/tex] [tex]x=\pm\sqrt{308+53}[/tex] [tex]x=\pm\sqrt{361}[/tex] [tex]x=\pm 9[/tex]