When x is divided by 3 remainder is z.In terms of z, which of the following could be equal to x?

Options

A. `z-3`
B. `3-z`
C. `3z`
D. `6+z`

Answer

To solve the problem, we need to express \( x \) in terms of \( z \) based on the information given about the division by 3 and the remainder.

### Step-by-Step Solution:

1. **Understanding the Division Rule**:
When a number \( x \) is divided by another number (in this case, 3), it can be expressed in the form:
\[ x = 3b + z \]
where:
- \( b \) is the quotient (which can be any integer),
- \( z \) is the remainder.

2. **Identifying the Remainder**:
The problem states that when \( x \) is divided by 3, the remainder is \( z \). According to the rules of division, the remainder \( z \) must satisfy the condition:
\[ 0 \leq z < 3 \]
This means \( z \) can be 0, 1, or 2.

3. **General Expression for \( x \)**:
From the division rule, we can express \( x \) as:
\[ x = 3b + z \]
Here \( b \) can take any integer value (0, 1, 2, ...).

4. **Finding Possible Values for \( x \)**:
Depending on the value of \( b \):
- If \( b = 0 \), then \( x = 0 + z = z \)
- If \( b = 1 \), then \( x = 3 + z \)
- If \( b = 2 \), then \( x = 6 + z \)
- If \( b = 3 \), then \( x = 9 + z \)
- And so on...

This means that \( x \) can take values such as \( z, 3 + z, 6 + z, 9 + z, \) etc.

5. **Conclusion**:
Therefore, in terms of \( z \), \( x \) can be expressed as:
\[ x = 3b + z \quad \text{for any integer } b \]
This means \( x \) can be any integer that, when divided by 3, leaves a remainder of \( z \).