**Solution:**

To find all numbers of the form 517xy that are divisible by 89, we need to analyze the divisibility criteria for 89 and apply it to the given number.

**Divisibility Criteria for 89:**

A number is divisible by 89 if and only if the difference between the number formed by its last two digits and the number formed by the remaining digits (excluding the last two digits) is divisible by 89.

**Step 1: Analyzing the given number**

The given number is in the form 517xy.

**Step 2: Forming the number for the last two digits**

Since the number is of the form 517xy, the number formed by the last two digits is xy.

**Step 3: Forming the number for the remaining digits**

To form the number for the remaining digits, we exclude the last two digits (xy) and consider the remaining digits, which are 517.

**Step 4: Applying the divisibility criteria**

According to the divisibility criteria for 89, the number 517 - xy must be divisible by 89 for the number 517xy to be divisible by 89.

**Step 5: Solving the equation**

To find the values of xy that make the equation 517 - xy divisible by 89, we can substitute different values for xy and check if the result is divisible by 89.

**Step 6: Finding the values of xy**

By substituting different values for xy, we find that the possible values for xy that make 517 - xy divisible by 89 are:

- xy = 28 (since 517 - 28 = 489, which is divisible by 89)
- xy = 61 (since 517 - 61 = 456, which is divisible by 89)

Therefore, the numbers of the form 517xy that are divisible by 89 are 51728 and 51761.

**Summary:**

The numbers of the form 517xy that are divisible by 89 are 51728 and 51761. These numbers satisfy the divisibility criteria for 89, which states that the difference between the number formed by the last two digits (xy) and the number formed by the remaining digits (517) must be divisible by 89. By substituting different values for xy, we find that only xy = 28 and xy = 61 make the equation 517 - xy divisible by 89.