When 93,119 and 158 are divided by the integer `n(n gt 1)`, the remainder in each case is d. What is the value of (n+d) ?
(a)17
(b)19
(c)14
(d)15

Options

A. 17
B. 19
C. 14
D. 15

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Understand the Problem
We need to find an integer `n` (where `n > 1`) such that when the numbers 93, 119, and 158 are divided by `n`, they all leave the same remainder `d`.

### Step 2: Calculate the Differences
To find a common divisor, we will calculate the differences between the numbers:
- Difference between 119 and 93:
\[ 119 - 93 = 26 \]
- Difference between 158 and 119:
\[ 158 - 119 = 39 \]
- Difference between 158 and 93:
\[ 158 - 93 = 65 \]

### Step 3: Find the HCF of the Differences
Now, we need to find the highest common factor (HCF) of the differences we calculated:
- The differences are 26, 39, and 65.

To find the HCF:
1. **Factors of 26**: 1, 2, 13, 26
2. **Factors of 39**: 1, 3, 13, 39
3. **Factors of 65**: 1, 5, 13, 65

The common factor among these is 13. Thus, the HCF is:
\[ \text{HCF}(26, 39, 65) = 13 \]

### Step 4: Assign the Value of n
From the HCF, we assign:
\[ n = 13 \]

### Step 5: Calculate the Remainder d
Now we need to find the remainder `d` when any of the numbers (let’s take 93) is divided by `n`:
\[ 93 \div 13 = 7 \quad \text{(quotient)} \]
Calculating the product:
\[ 13 \times 7 = 91 \]
Now, find the remainder:
\[ 93 - 91 = 2 \]
Thus, we have:
\[ d = 2 \]

### Step 6: Find n + d
Finally, we need to calculate:
\[ n + d = 13 + 2 = 15 \]

### Conclusion
The value of \( n + d \) is:
\[ \boxed{15} \]