Find the HCF using prime powers Compute the highest common factor (HCF) of 2^3, 3^2, and 15 by interpreting the compressed notation correctly.

Introduction / Context:
The raw stem “2 3, 3 2 and 15” is a compressed way to denote powers: 2^3 and 3^2, along with 15. We must carefully recover the intended values to apply HCF rules correctly.

Given Data / Assumptions:

• Numbers: 2^3 = 8, 3^2 = 9, and 15.
• All are positive integers.

Concept / Approach:
HCF requires the common prime factors with the smallest exponents across all numbers. If there is no prime common to all, the HCF is 1.

Step-by-Step Solution:
Prime factorisations: 8 = 2^3; 9 = 3^2; 15 = 3 * 5.Compare across all three: there is no prime that appears in all numbers (2 is missing from 9 and 15; 5 is missing from 8 and 9; 3 is missing from 8).Hence HCF = 1.

Verification / Alternative check:
Compute gcd sequentially: gcd(8, 9) = 1; then gcd(1, 15) = 1.

Why Other Options Are Wrong:
23 and 32 misread the notation as concatenation rather than powers; 360 is a product not a common factor; 3 is not a factor of 8.

Common Pitfalls:
Misinterpreting the compressed stem; assuming some non-existent shared factor; forgetting HCF must divide every number.

Final Answer:
1