Problem 13 - Elementary Algebra

Question

Find the greatest common factor. \(12 m^{2} n^{3}, 30 m^{5} n^{3}\)

Step-by-Step Solution

Answer

The GCF is 6m^2n^3.

Step 1: Find the Prime Factorization

Determine the prime factorization of each coefficient. For 12, the prime factorization is 12 = 2^2 * 3For 30, the prime factorization is 30 = 2 * 3 * 5

Step 2: Identify Common Factors in Coefficients

Identify the common prime factors from the factorizations found in Step 1. The common factors between 12 and 30 are 2 and 3.

Step 3: Determine the Lowest Powers of Common Prime Factors

Take the lowest powers of the common prime factors to form the GCF of the coefficients. The lowest power for 2 is 2^1, and for 3 it is 3^1.Hence, the GCF of the coefficients is 2^1 * 3^1 = 6

Step 4: Identify Common Variables and Their Lowest Exponents

Determine which variables are common in both terms and find the lowest exponent for each. Both terms contain the variables m and n. For m, the lowest exponent is m^2. For n, the lowest exponent is n^3.

Step 5: Combine GCF of Coefficients and Variables

Multiply the GCF of the coefficients with the lowest powers of the common variables. Thus, the GCF is 6 * m^2 * n^3.

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its basic building blocks: prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. For example, 12 can be factorized into prime numbers as 2 × 2 × 3, or in exponential form, 2^2 * 3. Similarly, 30 can be factorized as 2 × 3 × 5, or 2^1 * 3^1 * 5^1. Prime factorization helps in understanding the fundamental components of a number, which is very useful for finding the greatest common factor (GCF).

Common Factors

Common factors refer to the factors that two or more numbers share. In the context of our exercise, we first find the prime factorization of each number. Let's look at the numbers 12 and 30 again. The prime factorizations are:
12 = 2^2 * 3
30 = 2^1 * 3 * 5
To find the common factors, we identify the primes that appear in both factorizations. Here, both 12 and 30 have the prime numbers 2 and 3. These are the common factors, and we will use these to calculate the GCF, by taking the lowest power of each.

Lowest Exponents in Variables

When it comes to variables in algebra, the lowest exponent in common can also help find the GCF. Let's examine our variables, using the expression \(12m^2n^3\) and \(30m^5n^3\). Both terms have the variables m and n. For m, the lowest exponent is \(m^2\). For n, the lowest exponent is \(n^3\). By choosing the lowest exponents, we ensure that the common factors are as small as possible, which simplifies our greatest common factor.

Coefficients in Algebra

In algebra, coefficients are the numeric parts of terms that include variables. For example, in \(12m^2n^3\), 12 is the coefficient. Coefficients can be factorized just like whole numbers. First, factorize each coefficient into its prime factors:
12 = 2^2 * 3
30 = 2 * 3 * 5
Identify the lowest powers of the common prime factors. In this example, the lowest power for 2 is \(2^1\) and for 3 is \(3^1\). By multiplying these lowest powers, we get \(2^1 * 3^1 = 6\). This 6 represents the GCF for the coefficients.
Once we have the GCF of the coefficients and the variables, we combine them. So, the GCF of \(12m^2n^3\) and \(30m^5n^3\) results in:
6 * m^2 * n^3.