Question

Prove that if a ≡ b mod m and c ≡ d mod m, then ac ≡ bd mod m.

Answer 1

vvvvFinal answer:

To prove that if a ≡ b (mod m) and c ≡ d (mod m), then ac ≡ bd (mod m), consider the product ac - bd and use properties of modular arithmetic to show that it is divisible by m.

Step-by-step explanation:

We want to prove that if a ≡ b (mod m) and c ≡ d (mod m), then ac ≡ bd (mod m).

Since a ≡ b (mod m), it means that a - b is divisible by m.

Similarly, since c ≡ d (mod m), it means that c - d is divisible by m.

Now, let's consider the product ac - bd:

1. 
   
3. ac - bd = ac - ad + ad - bd
4. 
   
6. = a(c - d) + (a - b)d

Since (c - d) and (a - b) are both divisible by m, it follows that ac - bd is also divisible by m.

Therefore, we can conclude that ac ≡ bd (mod m).