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Chapter 10: Problem 17

Determine the greatest common factor. \(12,18,\) and 24

Short Answer

Expert verified
The greatest common factor is 6.

Step by step solution

01

Find the Prime Factors

Determine the prime factors of each number.Prime factors of 12: \[ 12 = 2 \times 2 \times 3 \] Prime factors of 18: \[ 18 = 2 \times 3 \times 3 \] Prime factors of 24: \[ 24 = 2 \times 2 \times 2 \times 3 \]
02

Identify Common Factors

Look for the prime factors that all three numbers have in common.Common prime factors: 2, 3
03

Calculate the Greatest Common Factor

Multiply the lowest powers of all common prime factors identified in Step 2.Lowest power of 2 in common: \(2^1\) Lowest power of 3 in common: \(3^1\) Therefore, the greatest common factor (GCF) is: \[ GCF = 2 \times 3 = 6 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Factors
Prime factors are the building blocks of numbers. When you break down a number into its prime factors, you reduce it to the smallest numbers that are only divisible by 1 and themselves. For example, for the number 12, the prime factors are 2 and 3. You can write it as: \(12 = 2 \times 2 \times 3\) Prime factorization helps to simplify numbers and make it easier to find commonalities with other numbers. This process is crucial when determining the greatest common factor (GCF).
Common Factors
Common factors are the numbers that are shared by two or more numbers. These factors are part of each of the original numbers. For instance, if we take the numbers 12, 18, and 24:
  • Prime factors of 12: \(2 \times 2 \times 3\)
  • Prime factors of 18: \(2 \times 3 \times 3\)
  • Prime factors of 24: \(2 \times 2 \times 2 \times 3\)
We see that 2 and 3 are common in all three sets of prime factors. Identifying these common factors allows us to later calculate the greatest common factor.
GCF Calculation
To calculate the Greatest Common Factor (GCF), we need to take the common factors identified previously and use their lowest powers. Here鈥檚 how:
  • Lowest power of 2 common to all numbers: \(2^1\)
  • Lowest power of 3 common to all numbers: \(3^1\)
By multiplying these, the GCF becomes: \(2 \times 3 = 6\). This step ensures we're using only the smallest necessary components to maintain the commonality while simplifying the calculation.
Step-by-Step Solutions
Let鈥檚 review the step-by-step solution for finding the GCF of 12, 18, and 24. Breaking it down into simple stages: 1. **Find Prime Factors:** Break down each number into its prime factors:
  • 12: \(2 \times 2 \times 3\)
  • 18: \(2 \times 3 \times 3\)
  • 24: \(2 \times 2 \times 2 \times 3\)
2. **Identify Common Factors:** Look for prime factors that all three numbers share鈥�2 and 3. 3. **Calculate the GCF:** Take the lowest powers of the common factors:
  • Lowest power of 2: \(2^1\)
  • Lowest power of 3: \(3^1\)
Multiply these together: \( 2 \times 3 = 6\).Congratulations! The greatest common factor of 12, 18, and 24 is indeed 6.

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Most popular questions from this chapter

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