Greatest divisor with given remainders: Find the largest integer that divides 130, 305, and 245 leaving remainders 6, 9, and 17 respectively.

Introduction / Context:When a divisor leaves different remainders with different numbers, subtracting the respective remainders from those numbers produces a set of integers all divisible by the same unknown divisor. The problem then reduces to computing the HCF of these adjusted numbers.

Given Data / Assumptions:

• N divides 130 with remainder 6 ⇒ N | (130 − 6) = 124.
• N divides 305 with remainder 9 ⇒ N | (305 − 9) = 296.
• N divides 245 with remainder 17 ⇒ N | (245 − 17) = 228.

Concept / Approach:The required divisor N is the HCF of 124, 296, and 228. Apply the Euclidean algorithm sequentially to find the HCF efficiently.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 5, 14, 24 are not common divisors of 124, 296, and 228; they fail at least one divisibility check.

Common Pitfalls:

• Mistaking the LCM approach; here we must use HCF after subtracting remainders.

Final Answer: