Prove or disprove that for any four integers a b c and d and

Prove or disprove that for any four integers a, b, c and d and any m > 1 if a b mod(m) and c d mod(m) then ac bd mod(m).

Solution

Proof:

Given a b (mod m) this states that a = b + qm

Given c d (mod m) this states that c = d + rm , some r,q

Consider ac – bd = (b + qm) ( d + rm) – bd

= bd + brm + qmd +qrm^2 – bd

= m (br + qd + qrm)

This shows that ac-bd is divisible by m,

so ac bd (mod m).

Hence the Proof.