Introduction:
In this problem, we are given two numbers, 511 and 667, and we need to find out how many numbers can be used as the divisor so that both numbers leave the same remainder when divided by that number.

Understanding the Problem:
To solve this problem, we need to find a common divisor for both numbers that will leave the same remainder when dividing both 511 and 667. In other words, we need to find a number that satisfies the equation:

511 ≡ 667 (mod n)

where "n" is the divisor.

Approach:
To find the common divisor, we can start by subtracting the two numbers and then finding the greatest common divisor (GCD) of the resulting difference. If the GCD is greater than 1, it means that there is a common divisor that satisfies the given condition.

Let's follow the steps to find the common divisor:

1. Calculate the difference between the two numbers:
Diff = 667 - 511 = 156

2. Find the GCD of the difference (156) and the smaller number (511) using Euclidean algorithm or any other method:
GCD(156, 511) = 1

3. Check if the GCD is greater than 1:
- If GCD > 1, there exists a common divisor that satisfies the given condition.
- If GCD = 1, there is no common divisor.

Result:
In this case, the GCD of the difference (156) and the smaller number (511) is 1, which means there is no common divisor greater than 1 that can be used to make both numbers leave the same remainder when divided by that number.

Conclusion:
Therefore, there are no numbers that can be used as the divisor to make 511 and 667 leave the same remainder when divided.

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