Question

Prove Congruence Property 3

C3 If a=b (mod m) and c < 1, then ac = bc (mod mc)

Answer 1

So, we have been given a = b (mod m) and we are asked to prove ac = bc (mod mc).

Or, we can also prove that ac - bc = 0 (mod mc).

Let's get to this.

Given, a = b (mod m)

We can write this as a - b = 0 (mod m).

Or, (a - b) = km ........................... (1)

Now, let's consider the left hand side of the equation we have to prove.

ac - bc = (a - b)c

We can substitute the value of (a - b) from equation (1).

We get ac - bc = kmc.

This means that (ac - bc) is divisible by mc.

So, ac - bc = 0 (mod mc)

Or, ac = bc (mod mc)

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