If the 8-digit number 7y9745x2 is divisible by 72, then the value of (2x-y) for the greatest value of x is:

Options

A. 18
B. 11
C. 14
D. 16

Answer

To determine the value of \(2x - y\) for the greatest value of \(x\) such that the 8-digit number \(7y9745x2\) is divisible by 72, we need to check the conditions for divisibility by both 8 and 9, since \(72 = 8 \times 9\).

### Step 1: Check divisibility by 8
A number is divisible by 8 if the last three digits form a number that is divisible by 8. In our case, the last three digits are \(x2\).

To check for divisibility by 8, we can express the last three digits as \(100x + 20\). We need to find values of \(x\) such that \(100x + 20\) is divisible by 8.

Calculating \(20 \mod 8\):
\[ 20 \div 8 = 2 \quad \text{(remainder 4)} \]
So,
\[ 20 \equiv 4 \mod 8 \]
This means we need:
\[ 100x + 20 \equiv 0 \mod 8 \implies 100x \equiv -4 \mod 8 \]
Calculating \(100 \mod 8\):
\[ 100 \div 8 = 12 \quad \text{(remainder 4)} \]
So,
\[ 100 \equiv 4 \mod 8 \]
Thus, we need:
\[ 4x \equiv -4 \mod 8 \implies 4x \equiv 4 \mod 8 \]
Dividing through by 4 gives:
\[ x \equiv 1 \mod 2 \]
This means \(x\) must be odd. The possible values for \(x\) are \(1, 3, 5, 7, 9\).

### Step 2: Check divisibility by 9
A number is divisible by 9 if the sum of its digits is divisible by 9. The digits of the number \(7y9745x2\) sum to:
\[ 7 + y + 9 + 7 + 4 + 5 + x + 2 = 34 + y + x \]
We need \(34 + y + x\) to be divisible by 9.

Calculating \(34 \mod 9\):
\[ 34 \div 9 = 3 \quad \text{(remainder 7)} \]
So,
\[ 34 \equiv 7 \mod 9 \]
Thus, we need:
\[ 7 + y + x \equiv 0 \mod 9 \implies y + x \equiv 2 \mod 9 \]

### Step 3: Finding values of \(x\) and \(y\)
Now we will substitute the possible odd values of \(x\) and check for corresponding \(y\) values.

1. **If \(x = 9\)**:
\[ y + 9 \equiv 2 \mod 9 \implies y \equiv -7 \equiv 2 \mod 9 \implies y = 2 \]
So, \(x = 9\) and \(y = 2\).

2. **If \(x = 7\)**:
\[ y + 7 \equiv 2 \mod 9 \implies y \equiv -5 \equiv 4 \mod 9 \implies y = 4 \]

3. **If \(x = 5\)**:
\[ y + 5 \equiv 2 \mod 9 \implies y \equiv -3 \equiv 6 \mod 9 \implies y = 6 \]

4. **If \(x = 3\)**:
\[ y + 3 \equiv 2 \mod 9 \implies y \equiv -1 \equiv 8 \mod 9 \implies y = 8 \]

5. **If \(x = 1\)**:
\[ y + 1 \equiv 2 \mod 9 \implies y \equiv 1 \mod 9 \implies y = 1 \]

### Step 4: Calculate \(2x - y\)
Now we calculate \(2x - y\) for the greatest value of \(x\) which is \(9\) and \(y = 2\):
\[ 2x - y = 2(9) - 2 = 18 - 2 = 16 \]

### Final Answer
The value of \(2x - y\) for the greatest value of \(x\) is:
\[ \boxed{16} \]