Multiple choice

The integers 34041 and 32506, when divided by a three-digit integer (n), leave the same remainder. What is the value of (n)?

  1. $289$
  2. $367$
  3. $453$
  4. $307$
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D Correct answer
Explanation

If both numbers leave the same remainder r when divided by n, then n divides their difference: 34041 - 32506 = 1535. Factorizing 1535 = 5 × 307. Testing divisors: 1535 ÷ 307 = 5, 1535 ÷ 453 ≈ 3.39. Both numbers must be > r, where r = 32506 mod n. For n=307: 32506 ÷ 307 = 105 remainder 271, 34041 ÷ 307 = 110 remainder 271. Same remainder confirms n=307.

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