Prove If a c mod n and b d mod n then ab cd mod n Let n Z

Prove If a = c mod n and b = d mod n then ab = cd mod n Let n Z prove that 3|2n^2 + 1 if and only if 3 does not divide n Extra Credit 123 = 6 mod 7 and 1 + 2 3 = 6 mod 7. 4498 = 4 mod 7 and 4 + 4 + 9 + 8 mod 7

Solution

As a= c mod n so a can be written as a = c + qn for some q

As b= d mod n so a can be written as b = d + rn for some r

ab – cd = (c + qn) ( d + rn) – cd = cd + crn + qnd +qrn^2 – cd = n (cr + qd + qrm) which is divisible by n, so

ab cd (mod n)

(Hence proved)