Difficulty: Easy
Correct Answer: The LCM is always less than or equal to the product of all the numbers.
Explanation:
Introduction / Context:
This is a conceptual question about the relationship between the Least Common Multiple (LCM) of a set of numbers and the product of those numbers. Rather than asking you to compute any specific numerical value, the question focuses on understanding how large or small the LCM can be when compared with the product. Concept questions like this are common in number system topics, because they help build intuition that can later be applied to trickier word problems and algebraic situations involving divisibility and multiples.
Given Data / Assumptions:
Concept / Approach:
The product of a set of positive integers includes every prime factor of each number, including repetitions. In contrast, the LCM uses the highest power of each distinct prime factor appearing in any of the numbers. This means that the LCM never has more prime factors (counted with multiplicity) than the product. As a result, the LCM is always less than or equal to the product. If the numbers are co prime, the LCM happens to be exactly equal to the product. If they share common factors, the LCM is strictly less than the product because repeated prime factors are not counted multiple times in the LCM.
Step-by-Step Solution:
Step 1: Recall definition of product.
Product = n1 * n2 * n3 * ... * nk.
Step 2: Recall definition of LCM.
LCM = smallest positive integer divisible by all n1, n2, ..., nk.
Step 3: Express each number in prime factor form.
Each number is written as a product of primes raised to some powers.
Step 4: For product, multiply all prime powers from all numbers.
Step 5: For LCM, take only the highest power of each distinct prime.
Step 6: Because LCM avoids unnecessary repetitions, it can never exceed the product and is therefore always less than or equal to the product.
Verification / Alternative check:
Consider a simple example with co prime numbers such as 2 and 3. Their product is 2 * 3 = 6. Their LCM is also 6 because 6 is the smallest number divisible by both 2 and 3. In this case, LCM equals product, which fits the statement that LCM is less than or equal to the product. Now take numbers that share a common factor, for example 4 and 6. The product is 4 * 6 = 24. The LCM is 12, which is clearly less than 24. Again, this supports the idea that LCM is less than or equal to the product, and sometimes strictly less.
Why Other Options Are Wrong:
The LCM is always greater than the product: This is false, because we just saw examples where LCM equals or is less than the product.
The LCM is always equal to the product: This is not true when numbers share common factors, such as 4 and 6, where LCM is 12 but product is 24.
The relationship between LCM and product cannot be determined: This is incorrect because number theory gives a clear relationship: LCM is always less than or equal to the product.
The LCM is always less than each individual number: This is false since LCM is at least as large as the largest of the given numbers, and often much larger.
Common Pitfalls:
Many learners confuse LCM with HCF and assume similar size relations without properly analyzing the definitions. Another frequent mistake is to think that LCM must always be equal to the product, mainly because in simple examples with co prime numbers that pattern holds. However, once numbers share factors, this assumption breaks down. Also, some students think the relationship is unpredictable, but in reality the prime factor method gives a very systematic way to understand the size of the LCM relative to the product.
Final Answer:
The minimum possible value of the LCM is always less than or equal to the product of all the given numbers.
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