When two positive integers A and B are divided by another number D, then the remainders obtained were 23 and 3 respectively. Further when `(A + B)` is divided by D, the remainder is 26, then value of D is :

Options

A. 62
B. 69
C. 26
D. Any value greater than 26

Answer

To solve the problem, we need to analyze the information given about the integers A and B when divided by D.

### Step-by-Step Solution:

1. **Understanding the Remainders:**
- When A is divided by D, the remainder is 23. This can be expressed as:
\[ A = kD + 23 \]
for some integer \( k \).
- When B is divided by D, the remainder is 3. This can be expressed as:
\[ B = mD + 3 \]
for some integer \( m \).

2. **Adding A and B:**
- Now, we need to find the sum \( A + B \):
\[ A + B = (kD + 23) + (mD + 3) = (k + m)D + (23 + 3) = (k + m)D + 26 \]

3. **Dividing A + B by D:**
- When \( A + B \) is divided by D, the remainder is given as 26. Therefore, we can write:
\[ A + B = nD + 26 \]
for some integer \( n \).

4. **Equating the Two Expressions for A + B:**
- From the two expressions for \( A + B \), we have:
\[ (k + m)D + 26 = nD + 26 \]
- By simplifying, we can eliminate 26 from both sides:
\[ (k + m)D = nD \]
- This implies:
\[ (k + m - n)D = 0 \]
- Since D cannot be zero, we conclude that:
\[ k + m = n \]

5. **Finding the Value of D:**
- Now, we know that the remainder when \( A + B \) is divided by D is 26. This means that D must be greater than 26, because the remainder must always be less than the divisor.
- Thus, the possible values for D are any integers greater than 26.

6. **Conclusion:**
- The value of D can be any integer greater than 26. Therefore, the correct answer is:
\[ \text{Option 4: Any value greater than 26} \]