If the 5-digit number 457 xy is divisible by 3,7 and 11, then what is the value of `(2x + 5y)` ?

Options

A. A)48
B. B)21
C. C)14
D. D)46

Answer

To solve the problem, we need to determine the values of \( x \) and \( y \) in the 5-digit number \( 457xy \) such that it is divisible by 3, 7, and 11. The product of these numbers is \( 231 \).

### Step-by-Step Solution:

1. **Understand the Divisibility Condition**:
The number \( 457xy \) must be divisible by \( 231 \).

2. **Calculate the Range of the Number**:
The number \( 457xy \) can range from \( 45700 \) to \( 45799 \).

3. **Find the Largest Multiple of 231 Less Than 45799**:
We divide \( 45799 \) by \( 231 \):
\[ 45799 \div 231 \approx 198.25 \]
Thus, the largest integer \( k \) such that \( 231k \) is less than \( 45799 \) is \( 198 \).

4. **Calculate \( 231 \times 198 \)**:
\[ 231 \times 198 = 45758 \]
So, \( 45758 \) is the largest multiple of \( 231 \) that is less than \( 45799 \).

5. **Calculate the Remainder**:
Now, we find the remainder when \( 45799 \) is divided by \( 231 \):
\[ 45799 - 45758 = 41 \]
Therefore, \( 457xy \) must be \( 45799 - 41 = 45758 \).

6. **Extract \( x \) and \( y \)**:
From \( 45758 \), we can see that \( x = 5 \) and \( y = 8 \).

7. **Calculate \( 2x + 5y \)**:
Now, we substitute \( x \) and \( y \) into the expression \( 2x + 5y \):
\[ 2x + 5y = 2(5) + 5(8) = 10 + 40 = 50 \]

### Final Answer:
The value of \( 2x + 5y \) is \( 50 \).