Final answer:
To prove that ac ≡ bd mod m, we can use the properties of congruence and algebraic manipulation. By expanding the expressions and combining like terms, we can show that ac - bd = nm(c - b) for some integer n. This implies that ac and bd have the same remainder when divided by m, proving the congruence.
Step-by-step explanation:
To prove that ac ≡ bd mod m, given a ≡ b mod m and c ≡ d mod m, we can use the properties of congruence. If a ≡ b mod m, then a-b is divisible by m. Similarly, if c ≡ d mod m, then c-d is divisible by m. Therefore, we can write a-b = km and c-d = lm for some integers k and l.
Multiplying both sides of the first equation by c and the second equation by b gives ac - bc = kmc and cb - bd = lbm.
Adding these two equations, we get ac - bc + cb - bd = kmc + lbm.
Combining like terms on both sides of the equation, we have ac - bd = km(c - b) + lm(c - b).
Factoring out (c - b) on the right side of the equation, we get ac - bd = (km + lm)(c - b).
Since km + lm is an integer, we can rewrite this as ac - bd = nm(c - b) for some integer n.
Therefore, we have proved that ac ≡ bd mod m.