Can the LCM of Two Integers Always Be Divided by Their GCD?

Many principles in mathematics are based on fundamental properties and definitions, lending them a certain inevitability. One such property is the relationship between the least common multiple (LCM) of two integers and their greatest common divisor (GCD). In this article, we will explore this relationship and provide a mathematical proof to demonstrate that the LCM of any two integers is always divisible by their GCD.

Introduction to LCM and GCD

To begin, let's revisit the definitions of LCM and GCD. The least common multiple (LCM) of two integers is defined as the smallest positive integer that is a multiple of both numbers. On the other hand, the greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest positive integer that divides both numbers without leaving a remainder.

Mathematical Proof of the LCM and GCD Relationship

Now, let's consider any two integers a and b. We can establish the relationship between their LCM and GCD using a fundamental equation:

LCM(a, b) * GCD(a, b)  |a * b|

By isolating LCM on one side of the equation, we obtain:

LCM(a, b)  |a * b| / GCD(a, b)

This expression clearly indicates that the LCM of any two integers is indeed divisible by their GCD. The division results in a quotient that is an integer, given that the numerator is always a multiple of the denominator. This holds true for all pairs of integers, regardless of their values.

Examples Demonstrating LCM and GCD Relationship

To further clarify the relationship between LCM and GCD, let's consider some examples:

Example 1: GCD and LCM with Co-prime Numbers

A set of numbers can have a GCD that is less than the individual numbers. For instance, consider the set {12, 18}. The GCD of 12 and 18 is 6, as both numbers are divisible by 6. The LCM of 12 and 18 is 36, which is a multiple of both numbers.

From these values, we can confirm the relationship as follows:

LCM(12, 18) * GCD(12, 18)  36 * 6  216  |12 * 18|

This example aligns with the mathematical proof, showing that the relationship holds true.

Example 2: LCM with Multiple of Co-prime Numbers

Consider another set of numbers, {14, 35}. Both 14 and 35 are not co-prime, as they share common factors. The GCD of 14 and 35 is 7, while the LCM is 70. We can verify the relationship as:

LCM(14, 35) * GCD(14, 35)  70 * 7  490  |14 * 35|

This example further reinforces the principle that the LCM of any two integers is divisible by their GCD.

Conclusion

In conclusion, the following mathematical principle stands true: the least common multiple (LCM) of two integers is always divisible by their greatest common divisor (GCD). This property is a fundamental aspect of number theory and is applicable to all integer pairs, regardless of their values.