When a number is divided by another number, the remainder is 23. When twice of this number is divided by the same divisor the remainder is 11. The divisor is—

Options

A. 34
B. 35
C. 20
D. 36

Answer

To solve the problem, let's break it down step by step.

### Step 1: Define the variables
Let the number be \( A \) and the divisor be \( D \).

### Step 2: Set up the first equation
According to the problem, when the number \( A \) is divided by \( D \), the remainder is 23. This can be expressed mathematically as:
\[ A = kD + 23 \]
where \( k \) is some integer (the quotient).

### Step 3: Set up the second equation
When twice the number \( 2A \) is divided by the same divisor \( D \), the remainder is 11. This can be expressed as:
\[ 2A = mD + 11 \]
where \( m \) is another integer (the new quotient).

### Step 4: Substitute the first equation into the second
From the first equation, we can express \( A \) in terms of \( D \):
\[ 2A = 2(kD + 23) = 2kD + 46 \]
Now substitute this into the second equation:
\[ 2kD + 46 = mD + 11 \]

### Step 5: Rearrange the equation
Rearranging gives us:
\[ 2kD + 46 - mD = 11 \]
This simplifies to:
\[ (2k - m)D + 46 = 11 \]
Now, isolate the term involving \( D \):
\[ (2k - m)D = 11 - 46 \]
\[ (2k - m)D = -35 \]

### Step 6: Solve for \( D \)
Since \( D \) must be a positive integer, we can express \( D \) as:
\[ D = \frac{-35}{2k - m} \]
To find a valid \( D \), \( 2k - m \) must be a negative divisor of -35. The possible divisors of -35 are: -1, -5, -7, -35, 1, 5, 7, 35.

### Step 7: Find valid values for \( D \)
We need \( D \) to be positive, so we consider the positive divisors of 35:
- If \( 2k - m = -1 \), then \( D = 35 \)
- If \( 2k - m = -5 \), then \( D = 7 \)
- If \( 2k - m = -7 \), then \( D = 5 \)
- If \( 2k - m = -35 \), then \( D = 1 \)

### Step 8: Check which divisor satisfies both conditions
We need to check which of these values for \( D \) satisfies the original conditions of the problem:
1. For \( D = 35 \):
- \( A = k \cdot 35 + 23 \)
- \( 2A = 2(k \cdot 35 + 23) = 70k + 46 \)
- The remainder when \( 70k + 46 \) is divided by 35 is \( 46 \mod 35 = 11 \) (satisfies condition).

Thus, the divisor \( D \) is **35**.

### Final Answer
The divisor is \( \boxed{35} \).

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