Problem 24 - Prealgebra

Question

Find the least common multiple (LCM) of each pair of numbers or monomials. $$20 c, 12 c$$

Step-by-Step Solution

Answer

The LCM of \(20c\) and \(12c\) is \(60c\).

Step 1: Understanding the Problem

We need to find the least common multiple (LCM) of the two monomials: \(20c\) and \(12c\). The LCM is the smallest expression that both monomials divide into without remainder.

Step 2: Factor Each Monomial

Factor the numerical coefficients and the variable parts separately: \( 20c = 2^2 \times 5 \times c \) \( 12c = 2^2 \times 3 \times c \).

Step 3: Identify the Highest Power for Each Factor

From the factorizations: - The highest power of 2 is \(2^2\). - The highest power of 3 is \(3^1\). - The highest power of 5 is \(5^1\). - The highest power of \(c\) is \(c^1\).

Step 4: Multiply the Highest Powers Together

Combine the highest powers of all factors to find the LCM: \( LCM = 2^2 \times 3^1 \times 5^1 \times c^1 = 60c \).

Step 5: Verify the LCM

Confirm that \(20c\) and \(12c\) divide evenly into \(60c\): - \(60c \div 20c = 3\) - \(60c \div 12c = 5\) This verifies that \(60c\) is the smallest multiple common to both \(20c\) and \(12c\).

Key Concepts

Understanding Monomials

Monomials are algebraic expressions consisting of a single term that can include numbers, variables, or the product of numbers and variables. They are simple building blocks in algebra, similar to words being the building blocks of sentences. For example, in the expression

• \(20c\), the term contains a numerical factor, 20, and a variable factor, \(c\).
• Similarly, in \(12c\), the number 12 is the numerical coefficient and \(c\) is the variable part.

Monomials can be added or multiplied. When combined, they follow the basic operations and rules of arithmetic and algebra. Understanding monomials is critical as it helps in operations involving polynomials and more complex algebraic expressions.
Learning to manage monomials prepares you for understanding more complicated expressions.

Explaining Factorization

Factorization is the process of breaking down numbers or expressions into a product of several factors. When factorizing monomials, you work on both the numerical coefficient and the variable parts separately.
Consider factorizing the monomials \(20c\) and \(12c\):

• For \(20c\): \(20\) factors into \(2^2 \times 5\), and the variable part is \(c\).
• For \(12c\): \(12\) factors into \(2^2 \times 3\), and the variable part is again \(c\).

This process helps in identifying each prime number and each variable component's contribution to the whole expression. Factorization is key to finding roots, simplifying expressions, and determining the least common multiple (LCM).

Importance of Numerical Coefficients

Numerical coefficients are the numbers that precede variables in terms. They are essential as they weigh the term with respect to the variable part. Consider \(20c\) and \(12c\) again:

• In \(20c\), 20 is the numerical coefficient weighing how many \(c\)'s we have.
• In \(12c\), 12 represents the same role but with a different weight.

Numerical coefficients determine the term's overall numerical value and influence the process of operations like addition, subtraction, and multiplication between terms. When finding the LCM, factorizing these coefficients gives insight into what multiplies to become the common multiple of the terms involved.

LCM Verification Process

After determining the least common multiple (LCM) of the given monomials, it is crucial to verify that the calculated LCM is indeed the smallest multiple common to both original terms. For the example of \(20c\) and \(12c\), we factorized them, then combined the highest powers of all contributing factors.
The calculated LCM, \(60c\), needs verification:

• Divide \(60c\) by \(20c\): The result is 3, showing \(20c\) divides evenly.
• Divide \(60c\) by \(12c\): The result is 5, showing \(12c\) also divides evenly.

This verification confirms our calculated LCM accurately represents the smallest common multiple. Ensuring LCM accuracy prevents errors in further calculations involving the monomials, aiding in problem-solving and ensuring consistency in results.