Compute 1355 Does the result of a show that 13 is not a squa

Compute (13/55) Does the result of (a) show that 13 is not a square mod 55? Show that 13 is not a square mod 5, and conclude that 13 is not a square mod 55.

Solution

a)

Let, x^2=13 mod 55

55=5*11

So,

x^2=13 mod 5=3 mod 5

But, squares modulo 5 give remainders: 0,1,-1

So, x^2=13 has no solution

HEnce

(13/55)=-1

b)

Yes

c)

We already proved this in part a)

And concluded that 13 is not a square mod 55