Question

There are two integers 34041 and 32506, when divided by a three-digit integer $n$, leave the same remainder. What is the value of $n$?

• A. 298
• B. 307
• C. 461
• D. can't be determined

Hint:

Same remainder $\Rightarrow$ divisor divides the difference. Then just check the admissible factors.

Answer 1

The Correct Option is B

Step 1: Every three-digit divisor of \( 1535 \) is a candidate, since a common remainder forces \( n \) to divide \( 34041-32506=1535 \).

Step 2: List the divisors of \( 1535 \) by testing its factor pairs: \[ 1535=1\times1535=5\times307. \] So the complete set of divisors is \( \{1,5,307,1535\} \).

Step 3: Among these, only \( 307 \) has exactly three digits, so it is the only possible value of \( n \).
\[ \boxed{n=307} \]