Question

Find the GCF using prime factorization. 130 and 78

Short answer

Answer: The GCF of 130 and 78 using the prime factorization method is 26.

Step-by-step solution

Find the prime factors of each number.

Firstly, we will find the prime factors of each number by dividing and breaking down each number into the smallest possible prime numbers. For 130: 130 ÷ 2 = 65 65 ÷ 5 = 13 So the prime factors of 130 are \(2 \times 5 \times 13\). For 78: 78 ÷ 2 = 39 39 ÷ 3 = 13 So the prime factors of 78 are \(2 \times 3 \times 13\).

Find the common prime factors.

Now, we must identify the prime factors that are common to both 130 and 78. These are the prime factors that occur in both sets of prime factors. The common prime factors are: \(2\) and \(13\).

Calculate the GCF.

Finally, to find the Greatest Common Factor (GCF), we multiply the common prime factors together. GCF = \(2 \times 13 = 26\) So, the Greatest Common Factor (GCF) of 130 and 78 using prime factorization is 26.

Key concepts

Prime Factorization

Prime factorization is the process of breaking down a composite number into its prime components, which when multiplied together, give back the original number. Think of it as finding the basic building blocks, or the prime numbers, that make up a more complex number.

For example, when looking at the number 130, we start by finding the smallest prime number that can divide it without leaving a remainder. We divide 130 by 2, the smallest prime number, resulting in 65. From there, we continue dividing by prime numbers until we're left with prime numbers only. With 130, we divided 65 by 5 to get 13, which is already a prime number. Hence, the prime factors of 130 are 2, 5, and 13.

GCF Calculation

To calculate the Greatest Common Factor (GCF), start by performing the prime factorization of each number involved. After finding the prime factors, the next step is to compare the lists of factors and identify which primes are common to all the numbers. The product of these common prime factors will be your GCF.

In our example with 130 and 78, we've found that the prime factors of 130 are 2, 5, and 13, while the prime factors of 78 are 2, 3, and 13. The common prime factors between these two sets are 2 and 13, as they appear in both lists. Multiplying these common factors, 2 and 13, we get the GCF, which is 26.

Common Prime Factors

Common prime factors are essential for finding the GCF. They are the prime numbers that appear in the prime factorization of each number you are comparing. It's like identifying similar ingredients in different recipes.

If we find 2 and 13 in the factorization of both 130 and 78, these numbers are our common prime factors. It's these shared prime numbers that play the pivotal role in GCF calculation, connecting the various numbers in your problem through their most basic components. Understanding common prime factors is crucial, as they form the heart of calculating not just the GCF, but also essential in many operations involving fractions, like simplifying or adding them.