For any integer n, evaluate HCF(22n + 7, 33n + 10).

Introduction: The HCF of linear expressions in n can often be simplified using the Euclidean algorithm by forming integer combinations that remove n, possibly yielding a constant HCF.

Given Data / Assumptions:

• Expressions: 22n + 7 and 33n + 10
• n is any integer

Concept / Approach: Apply the Euclidean algorithm to reduce the pair step by step until reaching a constant. If that constant is 1, the HCF is 1 for all integers n.

Step-by-Step Solution:

Verification / Alternative check: Since an integer combination of the two gives 1, they are coprime for all integer n.

Why Other Options Are Wrong: 0 is not a valid HCF here, 11 is not a universal divisor of both expressions for all n, and None of these does not apply because 1 is correct.

Common Pitfalls: Stopping the Euclidean steps early and missing the reduction to 1.

Final Answer: 1