Mathematical Principle

The problem is based on a fundamental property of modular arithmetic. If two integers, 'a' and 'b', leave the same remainder when divided by an integer 'n', then their difference (a - b) must be perfectly divisible by 'n'.

In this case, a = 34041, b = 32506, and 'n' is the 3-digit integer divisor.

Step 1: Calculate the Difference

According to the principle, the difference between the two numbers must be a multiple of 'n'.

Difference = 34041 - 32506 = 1535

This means that 'n' must be a factor of 1535.

Step 2: Find the Factors of 1535

To find the factors of 1535, we can perform prime factorization:

• The number ends in 5, so it is divisible by 5.
• 1535 ÷ 5 = 307

We then check if 307 is a prime number. We only need to test for divisibility by prime numbers up to $\sqrt{307}$ (which is approximately 17.5). Since 307 is not divisible by 2, 3, 5, 7, 11, 13, or 17, it is a prime number.

Therefore, the prime factorization of 1535 is 5 × 307.

The factors of 1535 are 1, 5, 307, and 1535.

Step 3: Identify the 3-Digit Divisor

The problem states that 'n' is a 3-digit integer. From the list of factors {1, 5, 307, 1535}, the only number that has three digits is 307.

Conclusion

The value of n must be 307.