When a certain number is divided by a certain divisor leaves remainder 43 and another no is divided by same divisor leaves remainder 37. Is sum of both number is divided by same divisor leaves remainder 13. find divisor .

Options

A. 65
B. 75
C. 77
D. 87

Answer

To solve the problem step by step, let's denote the divisor as \( x \).

### Step 1: Set up the equations based on the given information
When the first number is divided by \( x \), it leaves a remainder of 43. This can be expressed as:
\[ d_1 = k_1 \cdot x + 43 \]
where \( k_1 \) is some integer quotient.

When the second number is divided by \( x \), it leaves a remainder of 37:
\[ d_2 = k_2 \cdot x + 37 \]
where \( k_2 \) is another integer quotient.

### Step 2: Sum the two numbers
Now, we can sum both numbers:
\[ d_1 + d_2 = (k_1 \cdot x + 43) + (k_2 \cdot x + 37) \]
This simplifies to:
\[ d_1 + d_2 = (k_1 + k_2) \cdot x + 80 \]

### Step 3: Analyze the remainder when the sum is divided by \( x \)
According to the problem, when the sum \( d_1 + d_2 \) is divided by \( x \), it leaves a remainder of 13:
\[ d_1 + d_2 = m \cdot x + 13 \]
for some integer \( m \).

### Step 4: Set up the equation based on the remainders
From the two expressions for \( d_1 + d_2 \), we can equate the two:
\[ (k_1 + k_2) \cdot x + 80 = m \cdot x + 13 \]

### Step 5: Rearranging the equation
Rearranging gives us:
\[ (k_1 + k_2 - m) \cdot x = 13 - 80 \]
which simplifies to:
\[ (k_1 + k_2 - m) \cdot x = -67 \]

### Step 6: Finding the divisor \( x \)
Since \( x \) must be a positive divisor of -67, we can consider the positive divisors of 67. The divisors of 67 are 1 and 67.

However, since the remainders (43 and 37) must be less than \( x \), we can conclude:
- \( x \) must be greater than 43.

Thus, the only valid divisor is:
\[ x = 67 \]

### Conclusion
The divisor \( x \) is 67.