To solve the problem, we need to determine the values of `x` and `y` in the 10-digit number `897359y7x2` such that the number 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
To check for divisibility by 8, we need to look at the last three digits of the number, which are `7x2`. This means we need to find values of `x` such that `7x2` is divisible by 8.
**Hint:** The last three digits of a number determine its divisibility by 8.
- For `x = 1`: `712 ÷ 8 = 89` (divisible)
- For `x = 5`: `752 ÷ 8 = 94` (divisible)
- For `x = 9`: `792 ÷ 8 = 99` (divisible)
So, the possible values for `x` are 1, 5, and 9.
### Step 2: Check divisibility by 9
Next, we need to check the divisibility by 9. A number is divisible by 9 if the sum of its digits is divisible by 9. The sum of the digits in our number is:
\[ 8 + 9 + 7 + 3 + 5 + 9 + y + 7 + x + 2 = 50 + y + x \]
**Hint:** The sum of the digits must be divisible by 9.
We will check the values of `x` we found earlier (1, 5, and 9) and find the maximum possible value for `y`.
- For `x = 1`:
\[ 50 + y + 1 = 51 + y \]
To be divisible by 9, \( 51 + y \equiv 0 \mod 9 \).
\( 51 \mod 9 = 6 \), so \( y \equiv 3 \mod 9 \) → Possible values for `y`: 3, 12 (but `y` must be a single digit, so only 3).
- For `x = 5`:
\[ 50 + y + 5 = 55 + y \]
\( 55 \mod 9 = 1 \), so \( y \equiv 8 \mod 9 \) → Possible values for `y`: 8.
- For `x = 9`:
\[ 50 + y + 9 = 59 + y \]
\( 59 \mod 9 = 5 \), so \( y \equiv 4 \mod 9 \) → Possible values for `y`: 4.
### Step 3: Determine maximum value of `y`
From the calculations above, the possible values for `y` based on the values of `x` are:
- When `x = 1`, `y = 3`
- When `x = 5`, `y = 8`
- When `x = 9`, `y = 4`
The greatest value of `y` is 8 when `x` is 5.
### Step 4: Calculate \(3x - y\)
Now we can calculate \(3x - y\):
- \(x = 5\)
- \(y = 8\)
\[
3x - y = 3(5) - 8 = 15 - 8 = 7
\]
### Final Answer
The value of \(3x - y\) is **7**.
