If the 7- digit number 134x58y is divisible by 72, then the value of (2x + y) is

Options

A. 9
B. 6
C. 7
D. 8

Answer

To solve the problem, we need to determine the values of \( x \) and \( y \) in the 7-digit number \( 134x58y \) such that it is divisible by 72. A number is divisible by 72 if it is divisible by both 8 and 9.

### Step 1: Check divisibility by 8
For a number to be divisible by 8, the last three digits must form a number that is divisible by 8. The last three digits in our case are \( 58y \).

#### Calculation:
We will check the values of \( y \) from 0 to 9 to see which makes \( 58y \) divisible by 8.

- \( y = 0 \): \( 580 \div 8 = 72.5 \) (not divisible)
- \( y = 1 \): \( 581 \div 8 = 72.625 \) (not divisible)
- \( y = 2 \): \( 582 \div 8 = 72.75 \) (not divisible)
- \( y = 3 \): \( 583 \div 8 = 72.875 \) (not divisible)
- \( y = 4 \): \( 584 \div 8 = 73 \) (divisible)
- \( y = 5 \): \( 585 \div 8 = 73.125 \) (not divisible)
- \( y = 6 \): \( 586 \div 8 = 73.25 \) (not divisible)
- \( y = 7 \): \( 587 \div 8 = 73.375 \) (not divisible)
- \( y = 8 \): \( 588 \div 8 = 73.5 \) (not divisible)
- \( y = 9 \): \( 589 \div 8 = 73.625 \) (not divisible)

The only value of \( y \) that makes \( 58y \) divisible by 8 is \( y = 4 \).

### Step 2: Check divisibility by 9
Next, we need to check the sum of the digits \( 1 + 3 + 4 + x + 5 + 8 + 4 \) for divisibility by 9.

#### Calculation:
The sum of the digits is:
\[ 1 + 3 + 4 + x + 5 + 8 + 4 = 25 + x \]

We need \( 25 + x \) to be divisible by 9. Let's find the possible values of \( x \):

- \( 25 + 0 = 25 \) (not divisible)
- \( 25 + 1 = 26 \) (not divisible)
- \( 25 + 2 = 27 \) (divisible)
- \( 25 + 3 = 28 \) (not divisible)
- \( 25 + 4 = 29 \) (not divisible)
- \( 25 + 5 = 30 \) (not divisible)
- \( 25 + 6 = 31 \) (not divisible)
- \( 25 + 7 = 32 \) (not divisible)
- \( 25 + 8 = 33 \) (not divisible)
- \( 25 + 9 = 34 \) (not divisible)

The only value of \( x \) that makes \( 25 + x \) divisible by 9 is \( x = 2 \).

### Step 3: Calculate \( 2x + y \)
Now that we have \( x = 2 \) and \( y = 4 \), we can find \( 2x + y \):
\[ 2x + y = 2(2) + 4 = 4 + 4 = 8 \]

### Final Answer:
The value of \( 2x + y \) is \( 8 \).

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