A number when divided by a divisor leaves a remainder of 24. When twice the original number is divided by the same divisor, the remainder is 11. What is the value of divisor?

Options

A. 13
B. 59
C. 35
D. 37

Answer

To solve the problem step by step, we will denote the original number as \( n \) and the divisor as \( d \).

### Step 1: Set up the equations based on the problem statement.
From the problem, we know:
1. When \( n \) is divided by \( d \), the remainder is 24.
\[ n = d \cdot m + 24 \quad \text{(for some integer } m\text{)} \]

2. When \( 2n \) is divided by \( d \), the remainder is 11.
\[ 2n = d \cdot k + 11 \quad \text{(for some integer } k\text{)} \]

### Step 2: Substitute \( n \) from the first equation into the second equation.
Substituting \( n \) from the first equation into the second:
\[ 2n = 2(d \cdot m + 24) = 2dm + 48 \]
Now, we can rewrite the second equation:
\[ 2dm + 48 = d \cdot k + 11 \]

### Step 3: Rearrange the equation.
Rearranging gives:
\[ 2dm + 48 - 11 = d \cdot k \]
\[ 2dm + 37 = d \cdot k \]

### Step 4: Factor out \( d \).
This can be rewritten as:
\[ d \cdot (k - 2m) = 37 \]

### Step 5: Analyze the factors of 37.
Since 37 is a prime number, its only positive factors are 1 and 37. Therefore, \( d \) must be one of these factors:
1. If \( d = 1 \), then \( n \) would have to be 24, which does not satisfy the second condition (as \( 2n = 48 \) would give a remainder of 0 when divided by 1).
2. If \( d = 37 \):
- From the first equation:
\[ n = 37m + 24 \]
- From the second equation:
\[ 2n = 2(37m + 24) = 74m + 48 \]
When \( 74m + 48 \) is divided by 37, the remainder is:
\[ 74m + 48 \equiv 0m + 11 \quad (\text{since } 74 \equiv 0 \text{ and } 48 \equiv 11 \text{ mod } 37) \]
This satisfies the condition.

### Conclusion:
Thus, the value of the divisor \( d \) is \( 37 \).