Find the Least Common Multiple (LCM) of the five numbers 16, 24, 36, 52, and 54.

Introduction / Context:
This question asks for the Least Common Multiple, or LCM, of five numbers: 16, 24, 36, 52, and 54. Finding the LCM of many numbers is a classic test of prime factorization skills and an understanding of how multiples work. Problems like this appear in topics related to number systems, time schedules, and synchronization of events, where you must find a number that fits many different divisibility conditions at once.

Given Data / Assumptions:

• The numbers are 16, 24, 36, 52, and 54.
• All numbers are positive integers.
• We need to find their LCM.
• No additional constraints or modifications are mentioned.

Concept / Approach:
The most systematic way to find the LCM of several numbers is to use prime factorization. We express each number as a product of prime powers. Then we take the highest power of each prime that appears in any of the factorizations. Multiplying these highest powers gives the LCM. This method avoids mistakes that can happen if you try to guess or build up multiples manually, especially when dealing with many numbers at once.

Step-by-Step Solution:
Step 1: Prime factorize each number. 16 = 2^4. 24 = 2^3 * 3. 36 = 2^2 * 3^2. 52 = 2^2 * 13. 54 = 2 * 3^3. Step 2: List all distinct primes. The primes are 2, 3, and 13. Step 3: Take the highest power of each prime from the list. For 2, the highest power is 2^4 (from 16). For 3, the highest power is 3^3 (from 54). For 13, the highest power is 13^1 (from 52). Step 4: Multiply these highest powers. LCM = 2^4 * 3^3 * 13. Compute step by step: 2^4 = 16, 3^3 = 27. 16 * 27 = 432. 432 * 13 = 5616. Therefore, the LCM of 16, 24, 36, 52, and 54 is 5616.

Verification / Alternative check:
We can quickly check that each original number divides 5616. For example, 5616 / 16 = 351, which is an integer. 5616 / 24 = 234, 5616 / 36 = 156, 5616 / 52 = 108, and 5616 / 54 = 104. All these quotients are integers, so 5616 is a common multiple. To argue that it is the least such multiple, we rely on the prime factorization method, which by construction uses the minimum prime powers needed to cover all numbers. Any smaller number would omit at least one required prime factor or power and would fail to be divisible by all five numbers.

Why Other Options Are Wrong:
5618 and 5216: These numbers do not have the correct prime factor structure and fail to be exact multiples of all five given numbers simultaneously.
432: While 432 is a multiple of several of the numbers (such as 16 and 24), it is not a multiple of 52 and 54 together, so it cannot be the LCM.
3024: This is a common answer in many LCM problems but does not work here, because it is not divisible by 52 and does not represent the combined prime powers of all five numbers.

Common Pitfalls:
A typical mistake is to stop after computing the LCM of only some of the numbers instead of all of them. Another error is failing to take the highest power of a prime, leading to a smaller number that is not divisible by all the given numbers. Trying to list common multiples manually is also highly error prone with five numbers. Using organized prime factorization is the safest method and scales well when more numbers are involved.

Final Answer:
The LCM of 16, 24, 36, 52, and 54 is 5616.