# Proof again: Show that a mod m = b mod m if a = b (mod m)

## Homework Statement

Let m be a positive integer. Show that a mod m = b mod m if $a\,\equiv\,b\left(mod\,m\right)$.

## Homework Equations

Congruency - $$a\,\equiv\,b\left(mod\,m\right)$$

a is congruent to b modulo m.

## The Attempt at a Solution

I really, really, really suck at proofs. But here is what I tried.

$$a\,\equiv\,b\left(mod\,m\right)\,\longrightarrow\,a\,mod\,m\,\equiv\,\left[b\left(mod\,m\right)\right]\,\left(mod\,m\right)$$

matt grime
Homework Helper
Why are you trying to prove it using symbols?

Q. What does it mean in words for a to be congruent to b mod m?

Q. What is a mod m, and what is b mod m?

The definitions of these things mean there is *nothing* here that needs to be proved.

a is congruent to b mod m if m divides (a - b).

a mod m is the remainder of the division operation of $\frac{a}{m}$.

b mod m is the remainder of the division operation of $\frac{b}{m}$.

Does this count as "showing" a mod m $\equiv$ b mod m?

$$a\,\equiv\,b\left(mod\,m\right)\,\longrightarrow\,a\,mod\,m\,\equiv\,\left[b\left(mod\,m\right)\right]\,\left(mod\,m\right)$$

By definition of mod m and congruence mod m,

$$b(mod\,m)(mod\,m)\,\equiv\,b$$

So,

$$a(mod\,m)\,\equiv\,b$$

Then,

$$a\,\equiv\,\left[a(mod\,m)\right](mod\,m)\,\equiv\,a$$

Q.E.D.

matt grime
Homework Helper
Put it *simple* words. Phrases that involve the words: remainder on division by m. a=b mod m if and only if a and b have the same remainder on division by m. That's what modulo arithmetic. So, as you see, there is nothing to prove. a mod m is the remainder on division by m.

So all it is asking is: show that if a and b have the same remainder on division by m, then the remainder of a on division by m is the same as the remainder of b on division by m. So there isn nothing to prove beyond writing out the definitions.

a mod m is the remainder after a is divided by m.

a mod m and b mod m are equal if and only if their remainders are the same.

m divides (a - b) <---- What do I do with this?

Is that really it?

Last edited:
matt grime
Homework Helper
m divides (a - b) <---- What do I do with this?

nothing

Is that really it?
yes.

Review the definitions.

What is the mathematical definition of $a \equiv b \mbox{ (mod m)}$?