Find the largest number that divides 1305, 4665, and 6905 leaving the same remainder in each case. Also determine that common remainder.

Introduction / Context:Problems that ask for the largest number dividing several integers with the same remainder are classic Highest Common Factor (HCF) applications. The key insight is that if a number leaves the same remainder upon dividing multiple values, then it must exactly divide all pairwise differences of those values.

Given Data / Assumptions:

• Numbers: 1305, 4665, 6905
• Let the required divisor be D and the common remainder be r.
• Then 1305 = D*k1 + r, 4665 = D*k2 + r, 6905 = D*k3 + r for some integers k1, k2, k3.

Concept / Approach:The same remainder implies D divides pairwise differences. Hence D = HCF of {4665 - 1305, 6905 - 4665, 6905 - 1305}. After D is found, compute r from any original number: r = number - D * floor(number / D).

Step-by-Step Solution:

Verification / Alternative check:Check with 4665: 4665 / 1120 = 4 remainder 185 (since 1120*4 = 4480; 4665 - 4480 = 185). Check with 6905: 1120*6 = 6720; 6905 - 6720 = 185. Same remainder confirms correctness.

Why Other Options Are Wrong:

• 1210, 158: 1210 does not divide all differences 3360, 2240, 5600.
• 1120, 158: 1120 is correct but r = 185, not 158.
• 1210, 185: 1210 is not the HCF of the differences.
• None of these: A matching correct pair exists, so this is invalid.

Common Pitfalls:Using HCF of the original numbers instead of their differences, or computing the remainder with an incorrect dividend.

Final Answer:Largest divisor = 1120 and common remainder = 185.