For a given set of positive integers, how does the least common multiple (LCM) always compare to the product of all those numbers?

Introduction:
This conceptual number theory question tests your understanding of how the least common multiple (LCM) of several positive integers is related to the product of those integers. Understanding this relationship is crucial for quickly eliminating wrong options in LCM and HCF problems.

Given Data / Assumptions:

• We have a set of positive integers.
• We consider their least common multiple (LCM).
• We also consider the product of all these numbers.
• We need a statement that is always true when comparing LCM and the product.

Concept / Approach:
For any two positive integers a and b, we have the relationship:
a * b = HCF(a, b) * LCM(a, b)Since HCF(a, b) is at least 1, and usually greater than 1, this already suggests that LCM is less than or equal to the product. For more than two numbers, similar reasoning applies: common factors are shared between the numbers and counted fewer times in the LCM than in the full product.

Step-by-Step Solution:
Step 1: Consider two numbers a and b.We know: a * b = HCF * LCM.Step 2: Note that HCF(a, b) ≥ 1.So, LCM = (a * b) / HCF ≤ a * b.Step 3: Extend the idea to more numbers.When you have several integers, their LCM is formed by taking prime factors at the highest powers appearing in any one number, not multiplying all prime powers together as in the full product.Step 4: Conclusion.Therefore, the LCM of any collection of positive integers is always less than or equal to the product of all those integers.

Verification / Alternative check:
Example: numbers 4 and 6. Product = 24, LCM = 12, which is less than the product. For coprime numbers, such as 4 and 9, product = 36 and LCM = 36, which equals the product. In no case will the LCM exceed the full product.

Why Other Options Are Wrong:
Always greater than the product: impossible because LCM divides the product when considering all factors.
Always less than the product: false for coprime numbers where LCM equals the product.
Always equal to the product: false whenever numbers share common factors (like 4 and 6).
Cannot be determined in general: incorrect because we have a clear rule, less than or equal to the product.

Common Pitfalls:
Students often think LCM is always the product or confuse it with HCF. Another mistake is forgetting that for coprime numbers, LCM equals the product, not less.

Final Answer:
The correct comparison is that the LCM is always less than or equal to the product of the given numbers.