Problem 26 - Prealgebra

Question

Find the least common multiple (LCM) of each pair of numbers or monomials. $$7 x, 12 x$$

Step-by-Step Solution

Answer

The LCM of \(7x\) and \(12x\) is \(84x\).

Step 1: Identify the Numbers and Variables

We are given two monomials: \(7x\) and \(12x\). Our task is to find their least common multiple (LCM). First, observe the numerical coefficients and the variable part: \(7\), \(12\), and the common variable \(x\).

Step 2: Find the LCM of the Coefficients

The first step in finding the LCM of monomials is to determine the LCM of the numerical coefficients. Here the numbers are \(7\) and \(12\). - The prime factorization of \(7\) is \(7 = 7^1\).- The prime factorization of \(12\) is \(12 = 2^2 \times 3^1\).- To find the LCM, take the highest power of each prime present: - \(2^2 = 4\) - \(3^1 = 3\) - \(7^1 = 7\)- So, the LCM of \(7\) and \(12\) is \(2^2 \times 3^1 \times 7^1 = 84\).

Step 3: Consider the Variables

The monomials \(7x\) and \(12x\) both contain the variable \(x\). Since each monomial includes \(x^1\) and there are no additional powers of \(x\) in either term, we include \(x^1\) for the LCM.

Step 4: Compile the LCM of the Monomials

Combine the LCM of the coefficients and the variable part. The LCM of the coefficients is \(84\) and the variable part is \(x\). Thus, the LCM of the monomials \(7x\) and \(12x\) is \(84x\).

Key Concepts

Prime Factorization

Prime factorization is a method to express a number as a product of its prime factors. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. The process begins with dividing the given number by the smallest prime, 2, followed by the next smallest prime, 3, and so on, until the number reaches 1. This breakdown helps in determining many mathematical properties, like finding the least common multiple (LCM) or greatest common divisor (GCD). For instance, in the case of the numbers involved in our problem, the prime factorization of 7 is simple as 7 itself is a prime: \(7 = 7^1\). On the other hand, 12 can be broken down into \(12 = 2^2 \times 3^1\). Understanding the prime makeup makes it easier to find the LCM by taking the highest power of each prime from all numbers involved.

Monomials

A monomial is a mathematical expression that consists of a single term. It can be a constant number, a variable, or a product of numbers and variables raised to a power. Simply put, monomials have no addition or subtraction signs within them. Consider the monomials given in our problem: 7x and 12x. These each have a constant, 7 and 12, respectively, along with a variable, x. In the context of finding an LCM, we focus on both the coefficients and variables separately before combining them. Monomials with the same variables, like ours, make calculating the LCM more straightforward because we don't have to consider different variables interacting.

Variables

Variables act as placeholders for values that can change or vary, often represented using letters such as x, y, or z. They allow us to generalize mathematical statements and solutions. In the monomials 7x and 12x, the variable is x. When working with variables in monomials, if each monomial contains the same variable with equal power, as seen here with \(x^1\), the variable part of their LCM can straightforwardly include that common variable and its power. This consistency across monomials means that when determining the LCM, once the numerical part is resolved, the variable remains unchanged, yielding a combined expression like 84x.