To solve the problem, we need to find the values of \( x \) and \( y \) such that the 5-digit number \( 538xy \) is divisible by 3, 7, and 11.
### Step 1: Find the LCM of 3, 7, and 11
The first step is to find the least common multiple (LCM) of the numbers 3, 7, and 11. Since these numbers are all prime, the LCM is simply their product:
\[
\text{LCM} = 3 \times 7 \times 11 = 231
\]
**Hint:** To find the LCM of prime numbers, multiply them together.
### Step 2: Determine the range of the number
The number \( 538xy \) can be expressed as \( 53800 + 10x + y \). We need to find values of \( x \) and \( y \) such that this entire number is divisible by 231.
**Hint:** Express the number in a way that makes it easier to manipulate.
### Step 3: Check divisibility by 231
To find the largest 5-digit number \( 538xy \) that is divisible by 231, we can start with the largest possible value for \( xy \), which is 99 (i.e., \( x = 9 \) and \( y = 9 \)):
\[
53899 \div 231 \approx 233.4 \quad (\text{not an integer})
\]
Next, we check the next largest number, \( 53898 \):
\[
53898 \div 231 \approx 233.3 \quad (\text{not an integer})
\]
Continuing this process, we check \( 53897 \), \( 53896 \), and so on until we find a number that is divisible by 231.
**Hint:** Start from the largest possible combination and work downwards.
### Step 4: Find the largest divisible number
After checking several numbers, we find that \( 53862 \) is divisible by 231:
\[
53862 \div 231 = 233 \quad (\text{an integer})
\]
From this, we can see that \( x = 6 \) and \( y = 2 \).
**Hint:** Use division to check for divisibility and find the correct combination.
### Step 5: Calculate \( x^2 + y^2 \)
Now that we have \( x = 6 \) and \( y = 2 \), we can calculate \( x^2 + y^2 \):
\[
x^2 + y^2 = 6^2 + 2^2 = 36 + 4 = 40
\]
**Hint:** Remember to square each value and then add them together.
### Final Answer
Thus, the value of \( x^2 + y^2 \) is:
\[
\boxed{40}
\]
