Find the greatest number that, when dividing 390, 480, and 620, leaves the same remainder in each case.

Introduction / Context:When a single divisor leaves the same remainder upon division of several numbers, the differences between the numbers are exact multiples of that divisor. The largest such divisor equals the greatest common divisor (GCD) of all pairwise differences.

Given Data / Assumptions:

• Numbers: 390, 480, 620.
• One unknown divisor d leaves the same remainder on each division.

Concept / Approach:If 390 ≡ 480 ≡ 620 (mod d), then d divides 480 − 390, 620 − 480, and 620 − 390. So compute d = gcd(90, 140, 230). The largest such d is the answer.

Step-by-Step Solution:

Verification / Alternative check:Because 10 divides each difference, any common remainder r (0 ≤ r < 10) will be the same for all; no larger number divides all the differences simultaneously.

Why Other Options Are Wrong:

• 15, 20, 19, 30: These do not divide all of 90, 140, and 230 together; hence they cannot produce the same remainder for all three original numbers.

Common Pitfalls:

• Mistaking the GCD of the original numbers for the required divisor.
• Not considering all pairwise differences.

Final Answer:10