Repair and compute the LCM with powers Interpret the numbers as 3^3, 4, 4^2, and 3 (standard notation compressed in the source). Find the least common multiple (LCM) of 3^3, 4, 4^2, and 3.

Introduction / Context:
The raw stem showed compressed spacing: “3 3, 4, 4 2 and 3”. Applying the Recovery-First policy, we interpret this in standard exam style as powers: 3^3, 4, 4^2, and 3. We now compute the LCM of these integers.

Given Data / Assumptions:

• Numbers: 3^3 = 27, 4 = 2^2, 4^2 = 16 = 2^4, and 3 = 3^1.
• All are positive integers.

Concept / Approach:
LCM takes the highest power of each prime that appears in the factorisation across all numbers.

Step-by-Step Solution:
Prime 2: highest power among {2^2, 2^4} is 2^4 = 16.Prime 3: highest power among {3^3, 3^1} is 3^3 = 27.Therefore LCM = 2^4 * 3^3 = 16 * 27 = 432.

Verification / Alternative check:
432 ÷ 27 = 16, ÷ 16 = 27, ÷ 4 = 108, ÷ 3 = 144, all integers; any smaller candidate would miss at least one highest prime power.

Why Other Options Are Wrong:
12 and 48 do not contain sufficient powers (miss 3^3 or 2^4). “None of these” is invalid since 432 fits perfectly. 96 is still short of 3^3.

Common Pitfalls:
Adding exponents instead of taking maximum exponents, or misreading the compressed stem and losing the powers.

Final Answer:
432