When the integer n is divided by 2, the quotient is u and the remainder is 1. When the integer n is divided by 5, the quotient is v and the remainder is 3. Which one of the following must be true?

Options

A. `2u + 5v = 4`
B. `2u - 5v = 2`
C. `4u + 5v = 2`
D. `4u - 5v = 2`

Answer

To solve the problem, we need to express the integer \( n \) in terms of the quotients \( u \) and \( v \) based on the information given about the division by 2 and 5.

### Step-by-step Solution:

1. **Understanding the Division by 2:**
When \( n \) is divided by 2, the quotient is \( u \) and the remainder is 1. According to the division algorithm, we can express this as:
\[ n = 2u + 1 \]

2. **Understanding the Division by 5:**
When \( n \) is divided by 5, the quotient is \( v \) and the remainder is 3. Similarly, we can express this as:
\[ n = 5v + 3 \]

3. **Setting the Two Equations Equal:**
Since both expressions represent the same integer \( n \), we can set them equal to each other:
\[ 2u + 1 = 5v + 3 \]

4. **Rearranging the Equation:**
To find a relationship between \( u \) and \( v \), we can rearrange the equation:
\[ 2u + 1 - 3 = 5v \]
Simplifying this gives:
\[ 2u - 5v = 2 \]

5. **Final Result:**
Thus, we have derived the equation:
\[ 2u - 5v = 2 \]
This means that the correct option from the given choices is:
\[ \text{Option b: } 2u - 5v = 2 \]