To show that $a \mod m = b \mod m$ if $a \equiv b \mod m$, follow these steps:

Step 1: Definition of Congruence

The statement $a \equiv b \mod m$ means that $a$ and $b$ have the same remainder when divided by $m$. More formally, this means: $a - b = km \quad \text{for some integer } k.$ This implies that the difference between $a$ and $b$ is divisible by $m$.

Step 2: Modulo Operation Definition

The modulo operation $a \mod m$ gives the remainder when $a$ is divided by $m$. So, $a = mq_a + r_a \quad \text{and} \quad b = mq_b + r_b,$ where $r_a$ and $r_b$ are the remainders when $a$ and $b$ are divided by $m$, respectively.

Thus, $a \mod m = r_a \quad \text{and} \quad b \mod m = r_b.$

Step 3: Using Congruence to Show Equal Remainders

Since $a \equiv b \mod m$, we know that $a - b = km$ for some integer $k$. This implies that: $a = b + km.$ Now, consider the remainders of both $a$ and $b$ when divided by $m$. Since adding a multiple of $m$ (i.e., $km$) to a number does not affect its remainder when divided by $m$, we have: $a \mod m = b \mod m.$

Conclusion:

We have shown that if $a \equiv b \mod m$, then $a \mod m = b \mod m$. Therefore, the congruence of $a$ and $b$ modulo $m$ implies that they have the same remainder when divided by $m$.

Would you like more details or clarification on any part of this?

Here are 5 related questions to expand on this topic:

1. How does the division algorithm help in understanding modular arithmetic?
2. Can you generalize this result to other moduli besides integers?
3. How does the property $a \equiv b \mod m$ help in solving Diophantine equations?
4. What is the relationship between congruence and equivalence classes in modular arithmetic?
5. Can we extend this result to systems of congruences (e.g., Chinese Remainder Theorem)?

Tip: When working with modular arithmetic, remember that congruence relations behave similarly to equality in many algebraic operations (e.g., addition, subtraction, multiplication).