If the 10-digit number `897359y7x2` is divisible by 72, then what is the value of `(3x+y)`, for the possible greatest value of y?

Options

A. 31
B. 28
C. 27
D. 25

Answer

To solve the problem, we need to determine the values of \(x\) and \(y\) such that the 10-digit number \(897359y7x2\) 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 of the number must be divisible by 8. The last three digits in our case are \(7x2\).

To find the values of \(x\) that make \(7x2\) divisible by 8, we can check each digit from 0 to 9:

- \(x = 0\): \(702 \div 8 = 87.75\) (not divisible)
- \(x = 1\): \(712 \div 8 = 89.00\) (divisible)
- \(x = 2\): \(722 \div 8 = 90.25\) (not divisible)
- \(x = 3\): \(732 \div 8 = 91.50\) (not divisible)
- \(x = 4\): \(742 \div 8 = 92.75\) (not divisible)
- \(x = 5\): \(752 \div 8 = 94.00\) (divisible)
- \(x = 6\): \(762 \div 8 = 95.25\) (not divisible)
- \(x = 7\): \(772 \div 8 = 96.50\) (not divisible)
- \(x = 8\): \(782 \div 8 = 97.75\) (not divisible)
- \(x = 9\): \(792 \div 8 = 99.00\) (divisible)

The possible values for \(x\) that make \(7x2\) divisible by 8 are \(1, 5, 9\).

### Step 2: Check divisibility by 9
Next, we need to ensure that the sum of all the digits of the number \(897359y7x2\) is divisible by 9.

The sum of the known digits is:
\[ 8 + 9 + 7 + 3 + 5 + 9 + 7 + 2 = 50 \]
Including \(y\) and \(x\), the total sum becomes:
\[ 50 + y + x \]
We need \(50 + y + x\) to be divisible by 9.

Now we will check each possible \(x\) value:

1. **For \(x = 1\)**:
\[ 50 + y + 1 = 51 + y \]
We need \(51 + y\) to be divisible by 9. The smallest \(y\) that satisfies this is \(6\) (since \(51 + 6 = 57\) which is divisible by 9).

2. **For \(x = 5\)**:
\[ 50 + y + 5 = 55 + y \]
We need \(55 + y\) to be divisible by 9. The smallest \(y\) that satisfies this is \(1\) (since \(55 + 1 = 56\) which is not divisible, but \(55 + 4 = 59\) is also not divisible, but \(55 + 8 = 63\) is divisible).

3. **For \(x = 9\)**:
\[ 50 + y + 9 = 59 + y \]
We need \(59 + y\) to be divisible by 9. The smallest \(y\) that satisfies this is \(5\) (since \(59 + 5 = 64\) is not divisible, but \(59 + 8 = 67\) is also not divisible, but \(59 + 4 = 63\) is divisible).

### Step 3: Find the greatest value of \(y\)
The possible pairs \((x, y)\) from the above calculations are:
- \( (1, 6) \)
- \( (5, 1) \)
- \( (9, 5) \)

The greatest value of \(y\) is \(6\) when \(x = 1\).

### Step 4: Calculate \(3x + y\)
Now we can calculate \(3x + y\):
\[ 3x + y = 3(1) + 6 = 3 + 6 = 9 \]

### Final Answer
The value of \(3x + y\) for the possible greatest value of \(y\) is \(9\).