If two numbers are each divided by the same divisor, the remainders are respectively 3 and 4. If the sum of the two numbers be divided by the same divisor, the remainder is 2. The divisor is

Options

A. 9
B. 7
C. 5
D. 3

Answer

To solve the problem step by step, let's denote the two numbers as \( n_1 \) and \( n_2 \), and let the divisor be \( d \).

### Step 1: Set up the equations based on the remainders
From the problem, we know:
- When \( n_1 \) is divided by \( d \), the remainder is 3. This can be expressed as:
\[ n_1 = kd + 3 \quad \text{(for some integer } k\text{)} \]
- When \( n_2 \) is divided by \( d \), the remainder is 4. This can be expressed as:
\[ n_2 = md + 4 \quad \text{(for some integer } m\text{)} \]

### Step 2: Express the sum of the two numbers
Now, we can find the sum of the two numbers:
\[ n_1 + n_2 = (kd + 3) + (md + 4) = (k + m)d + 7 \]

### Step 3: Analyze the remainder when the sum is divided by the divisor
According to the problem, when the sum \( n_1 + n_2 \) is divided by \( d \), the remainder is 2. Therefore:
\[ (k + m)d + 7 \equiv 2 \mod d \]
This implies:
\[ 7 \equiv 2 \mod d \]

### Step 4: Simplify the congruence
To find \( d \), we can rearrange the above congruence:
\[ 7 - 2 \equiv 0 \mod d \implies 5 \equiv 0 \mod d \]
This means that \( d \) must be a divisor of 5.

### Step 5: Identify the possible values of the divisor
The divisors of 5 are 1 and 5. However, since the remainders given in the problem (3 and 4) must be less than the divisor, we can eliminate 1 as a possible divisor.

### Conclusion
Thus, the only possible value for the divisor \( d \) is:
\[ d = 5 \]

### Final Answer
The divisor is \( \boxed{5} \).