Question

The least common multiple (LCM) of two positive integers \(x\) and \(y\) is the smallest positive integer \(z\) such that \(z\) is an integer multiple of both \(x\) and \(y\). Give a formula for the least common multiple that involves the GCD.

Short answer

The LCM of two numbers \(x\) and \(y\) is \( \frac{x \times y}{\text{GCD}(x, y)} \).

Step-by-step solution

Understanding GCD

The greatest common divisor (GCD) of two numbers is the largest positive integer that divides both numbers without leaving a remainder. Finding the GCD of two numbers, say \(x\) and \(y\), helps us understand their common factors.

Prime Factorization

Both the LCM and GCD concepts rely on prime factorization. If \(x\) and \(y\) are expressed in terms of their prime factors, the GCD uses the lowest power of each common prime while the LCM uses the highest power of each prime present in either number.

Connection Between LCM and GCD

The product of the LCM and GCD of two numbers is equal to the product of the numbers: \( \text{LCM}(x, y) \times \text{GCD}(x, y) = x \times y \). This results from the way prime factors are distributed in LCM and GCD.

Deriving the Formula for LCM

Using the relationship \( \text{LCM}(x, y) \times \text{GCD}(x, y) = x \times y \), we can derive the formula for LCM as: \[ \text{LCM}(x, y) = \frac{x \times y}{\text{GCD}(x, y)} \]. This formula utilizes both the GCD and the product of the numbers to find the smallest number divisible by both.

Key concepts

Greatest Common Divisor (GCD)

The Greatest Common Divisor, abbreviated as GCD, plays an essential role in various mathematical computations. The GCD of two integers is the greatest positive integer that precisely divides each of the numbers, without leaving a remainder. This makes the GCD a handy tool in simplifying fractions and understanding the numbers' divisibility patterns.

To find the GCD of two numbers, such as 18 and 24, we can list their divisors or utilize the Euclidean algorithm, which iteratively applies the division process. This method is highly efficient, especially for larger numbers.

• Start with both numbers: 18 and 24.
• Divide the larger number by the smaller number: 24 divided by 18 gives a remainder of 6.
• Replace the larger number with the remainder, and repeat: use 18 and 6.
• Continue until a remainder of 0 is reached. The divisor at this stage, 6, is the GCD.

This algorithm helps us quickly find the GCD, paving the way for further calculations like finding the Least Common Multiple.

Prime Factorization

Prime factorization is a critical step in finding the GCD and LCM of numbers. It involves expressing a number as a product of its prime factors, which are numbers greater than 1 that have no divisors other than 1 and themselves. For example, the prime factorization of 18 is 2 × 3 × 3, while that of 24 is 2 × 2 × 2 × 3.

This process reveals the number's building blocks. For the GCD, we take the lowest power of each common prime factor present in both factorizations. In contrast, for the LCM, we use the highest power of each prime number found in either number's prime factorization.

• GCD of 18 and 24: Take the smallest power of common primes: 2 and 3. So, GCD is 2 × 3 = 6.
• LCM of 18 and 24: Take the largest power of all prime factors: 2³ and 3². Hence, LCM is 72.

Understanding prime factorization makes it easier to tackle more complex problems by breaking them down into manageable parts.

Mathematical Formulas

Mathematical formulas serve as precise and efficient tools to solve numerical problems, such as calculating the Least Common Multiple (LCM) using the Greatest Common Divisor (GCD). The relationship between the LCM and GCD of two numbers is elegantly captured in the formula: \[\text{LCM}(x, y) = \frac{x \times y}{\text{GCD}(x, y)}\] This formula reveals that the product of a number's LCM and GCD equals the product of the two numbers themselves. This interconnection reflects the manner in which the prime factors, specifically their exponents, are distributed between the GCD and LCM.

Applying the formula requires prior calculation or knowledge of the two given numbers' GCD. Once the GCD is determined, calculating the LCM becomes straightforward by multiplying the numbers and dividing by their GCD. This method is not only efficient but also serves as a nifty check: if any errors slip into prior calculations, this formula helps ensure accuracy. Understanding and employing such mathematical formulas is crucial for students in efficiently solving mathematical problems.