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

If two numbers $a$ and $b$ leave the same remainder on division by $n$, then $n$ divides their difference. \[ a-b=34041-32506=1535. \] Thus $n$ must be a three-digit divisor of $1535$. Factorize: \[ 1535=5\times 307. \] The only three-digit divisor is $307$ (since $5$ is one digit and $1535$ itself is four digits). Hence $n=307$. 

Answer 2

Since both numbers leave the same remainder on division by \( n \), each candidate value of \( n \) can be tested directly by computing the remainders of \( 34041 \) and \( 32506 \).

1. Option A (298): \[ 34041 = 298\times114+69, \qquad 32506=298\times109+24. \] The remainders \( 69 \) and \( 24 \) are different, so \( n=298 \) is ruled out.
2. Option B (307): \[ 34041=307\times110+271, \qquad 32506=307\times105+271. \] Both remainders equal \( 271 \), so \( n=307 \) satisfies the condition.
3. Option C (461): \[ 34041=461\times73+388, \qquad 32506=461\times70+236. \] The remainders \( 388 \) and \( 236 \) differ, so \( n=461 \) is ruled out.
4. Option D (can't be determined): Since one value of \( n \), namely \( 307 \), has already been shown to satisfy the given condition exactly, the value is determined, so this option cannot hold.

Only \( n=307 \) makes the two remainders equal, confirming option B.

Hence, the correct answer is 307.