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

Let n=pq, with pq and p, q are odd prime if a^2 b^2(mod n), ab(mod n) and a-b(mod n) then either (a+b,n) =p or (a+b,n) = q

Solution: - Consider a^2 b^2(mod n)

a^2-b^2=nk

(a-b)(a+b)=nk

(a-b)(a+b)=pqk

Thus, gcd of a+b and n will wither be equal to p or q.

Hence,if a^2 b^2(mod n), ab(mod n) and a-b(mod n) then either (a+b,n) =p or (a+b,n) = q

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

Solution

correct