If two numbers when divisible by a certain divisor give remainder 35 and 30 respectively and when their sum is divided by the same divisor, the remainder is 20, then the divisor is

Options

A. 40
B. 45
C. 50
D. 55

Answer

To solve the problem step by step, we can follow this approach:

### Step 1: Define the variables
Let the divisor be \( X \). The two numbers can be expressed in terms of \( X \) and their respective remainders.

### Step 2: Express the two numbers
1. The first number can be expressed as:
\[ \text{First number} = aX + 35 \]
where \( a \) is the quotient when the first number is divided by \( X \).

2. The second number can be expressed as:
\[ \text{Second number} = bX + 30 \]
where \( b \) is the quotient when the second number is divided by \( X \).

### Step 3: Find the sum of the two numbers
The sum of the two numbers is:
\[ \text{Sum} = (aX + 35) + (bX + 30) = (a + b)X + 65 \]

### Step 4: Analyze the remainder when the sum is divided by \( X \)
According to the problem, when the sum is divided by \( X \), the remainder is 20. Therefore, we can write:
\[ (a + b)X + 65 \equiv 20 \mod X \]
This implies that the term \( (a + b)X \) is divisible by \( X \), and we only need to consider the remainder of 65 when divided by \( X \):
\[ 65 \equiv 20 \mod X \]

### Step 5: Set up the equation
From the above, we can set up the equation:
\[ 65 - 20 = kX \quad \text{for some integer } k \]
This simplifies to:
\[ 45 = kX \]

### Step 6: Solve for \( X \)
Since \( X \) must be a divisor of 45, we can find the possible values of \( X \). The divisors of 45 are 1, 3, 5, 9, 15, and 45.

### Step 7: Determine the valid divisor
Since the remainders given in the problem (35 and 30) must be less than \( X \), the only possible value for \( X \) that satisfies both conditions (i.e., \( X > 35 \) and \( X > 30 \)) is:
\[ X = 45 \]

### Conclusion
Thus, the divisor is:
\[ \boxed{45} \]