Determine is in general true or false Recall that a universa

Determine is, in general, true or false. Recall that a
universal statement is true if it is true for all possible cases while it is false if there is even one
counterexample. Be prepared to prove that your answer is correct by supplying a proof or
counterexample, whichever is appropriate

If (a/b)=-1 and (b/q)=-1 then x^2 ab (mod pq) where p and q are odd prime has a solution.

solution:

(a/b)=-1

a^((p-1)/2) -1 (mod p)

a^((p-1)/2) +1 =pk

(b/q)=-1

b^((q-1)/2) -1 (mod q)

b^((q-1)/2) +1 =qk

Now multiplying both the equation we get

[a^((p-1)/2) +1] [b^((q-1)/2) +1 ]=pqk\'

(ab)^(pq-1)/2 -1=pqk\'

x^2ab (mod pq) has solution

Could you check it for me please is it correct or not?

Solution

yes it is correct