Question

Notes 2.7 Using CRT notation, show what is going on for all the combinations you considered in Notes 2.6. Explain why gcd(s + t, 35) sometimes gave you a factor, and it sometimes did not

Notes 2.6 is:

Notes 2.6 Consider all the possible sets of two square roots s, t of 1 (mod 35) where s ≢ t (mod 35) (there are six of them, since addition is commutative (mod 35).

For all possible combinations, compute gcd(s + t, 35). Which combinations give you a single prime factor of 35?

Answer 1

here i am use the definition of CRT and solve up to last lastly we can see that if s divides t then the gcd of 35 and sum of s and t is not a prime factor of 35 in fact it is multiple of 35