Two numbers A and B are both divisible by x. The difference A-B is divisible by x

Answer

To prove that if two numbers A and B are both divisible by x, then their difference A - B is also divisible by x, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Divisibility**:
- Since A is divisible by x, we can express A as:
\[ A = a \cdot x \]
where \( a \) is some integer.

2. **Expressing B**:
- Similarly, since B is also divisible by x, we can express B as:
\[ B = b \cdot x \]
where \( b \) is another integer.

3. **Finding the Difference**:
- Now, we need to find the difference A - B:
\[ A - B = (a \cdot x) - (b \cdot x) \]

4. **Factoring Out x**:
- We can factor out x from the expression:
\[ A - B = (a - b) \cdot x \]

5. **Conclusion**:
- Since \( (a - b) \) is an integer (the difference of two integers is an integer), we can conclude that:
\[ A - B \text{ is divisible by } x \]
- Therefore, we have proved that the difference A - B is also divisible by x.