Reconstruct the divisor from long-division remainders In a long division, the dividend is 380606 and the successive remainders (first to last) are 434, 125, and 413. What is the divisor?

Introduction / Context:
In multi-digit long division, each partial remainder is strictly less than the divisor, and the final remainder relates the dividend, quotient, and divisor. Sometimes, recognizing compatible quotient–divisor pairs solves the puzzle rapidly.

Given Data / Assumptions:

• Dividend N = 380606.
• Successive remainders: 434 → 125 → 413 (final).
• Remainders must be less than the divisor.

Concept / Approach:
Try the plausible three-digit divisor exceeding the observed remainders. Check whether N can be expressed as divisor * some integer + final remainder. If an option also matches a natural quotient seen in long division, it is very likely correct.

Step-by-Step Solution:
Test divisor d = 843 (greater than 434, 125, 413).Compute a trial quotient: 380606 − 413 = 380193.Check 380193 ÷ 843 = 451 exactly (since 843 × 451 = 380193).Hence N = 843 × 451 + 413, consistent with the given final remainder and the long-division structure.

Verification / Alternative check:
The presence of 451 among options hints it is the quotient for d = 843. Moreover, all intermediate remainders are less than 843, which is necessary for validity.

Why Other Options Are Wrong:
451 is a quotient, not a divisor; 4215, 3372 are unreasonably large and do not align cleanly with the final remainder structure.

Common Pitfalls:
Forgetting that remainders must be less than the divisor; trying to compute every long-division step instead of verifying N = d*q + r_final.

Final Answer:
843