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 $n=307$.