Let x be the greatest number by which when 448. 678 and 908 are divided ,the remainder in each case is 11 . When 147 is divided by x , then remainder is :

Options

A. 4
B. 5
C. 9
D. 3

Answer

To solve the problem step-by-step, we will find the greatest number \( x \) such that when 448, 678, and 908 are divided by \( x \), the remainder is 11. Then, we will find the remainder when 147 is divided by \( x \).

### Step 1: Subtract the remainder from each number
To find \( x \), we first need to subtract the remainder (11) from each of the given numbers:

- \( 448 - 11 = 437 \)
- \( 678 - 11 = 667 \)
- \( 908 - 11 = 897 \)

### Step 2: Find the HCF of the resulting numbers
Next, we need to find the highest common factor (HCF) of the numbers 437, 667, and 897.

1. **Factorization of 437**:
- 437 can be factored as \( 19 \times 23 \).

2. **Factorization of 667**:
- 667 can be factored as \( 29 \times 23 \).

3. **Factorization of 897**:
- 897 can be factored as \( 39 \times 23 \).

### Step 3: Identify the common factors
From the factorizations, we can see that the common factor among all three numbers is \( 23 \). Therefore, the HCF is:

\[ x = 23 \]

### Step 4: Find the remainder when 147 is divided by \( x \)
Now, we need to find the remainder when 147 is divided by \( x \):

\[ 147 \div 23 \]

Calculating this gives:

- \( 23 \times 6 = 138 \)

Now, subtract this product from 147 to find the remainder:

\[ 147 - 138 = 9 \]

### Final Answer
Thus, the remainder when 147 is divided by \( x \) (which is 23) is:

\[ \text{Remainder} = 9 \]