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Chapter 5: Problem 6

Find the greatest common factor. $$6,12,21$$

Short Answer

Expert verified
The greatest common factor of 6, 12 and 21 is 3.

Step by step solution

01

Prime Factorization of Each Number

Find the prime factors of each number. The prime factors of 6 are 2 and 3, the prime factors of 12 are 2, 2, and 3, and the prime factors of 21 are 3 and 7.
02

Find Common Prime Factors

Identify the common prime factors across the three numbers. The number 3 appears in all three lists, and 2 appears in the first two lists. Hence, 3 is a common factor.
03

Calculating the GCF

The GCF is the product of the lowest powers of all common factors. The only common factor is 3 to the power 1, hence GCF is 3.

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Key Concepts

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

Prime Factorization
Prime factorization is an important step in finding the greatest common factor (GCF) of a set of numbers. It involves breaking down each number into its prime numbers, which are numbers greater than 1 that have only two divisors: 1 and themselves. For instance, in the number 6, the prime factors are 2 and 3, because 2 multiplied by 3 equals 6.
Here's a deeper look at the process:
  • 6: Break it down into the prime numbers 2 and 3.
  • 12: It can be expressed as 2 times 2 times 3, or in exponential form, 2虏 脳 3.
  • 21: This number breaks down into 3 and 7.
Knowing the prime factors of each number allows you to easily find common factors and complete the GCF calculation.
Common Factors
Once you have the prime factors, the next step is to find the common factors among the numbers. A common factor is a number that divides exactly into two or more numbers without leaving a remainder. In our example, the numbers 6, 12, and 21, we look through their prime factors to find these common elements.
Here's how you identify them:
  • Check for any primes that appear in all the lists. Here, 3 is present in each of the sets: 6 (2 脳 3), 12 (2虏 脳 3), and 21 (3 脳 7).
  • While 2 appears twice in the factorization of 12, it's not in the factors of 21, which means it鈥檚 not a common factor across 6, 12, and 21.
Therefore, the only common factor in the given numbers is 3, marking a crucial part in finding the greatest common factor.
GCF Calculation
The final step involves determining the greatest common factor using the identified common factors. The GCF is the largest number that can divide each of the numbers completely. To find it, multiply the common prime factors using their lowest powers that appear in all numbers.

In the case of 6, 12, and 21:
  • The number 3 is the only common factor across all numbers.
  • The lowest power of 3 found in each number鈥檚 factorization is 3 to the power of 1 (3鹿).
Thus, the greatest common factor (GCF) of 6, 12, and 21 is simply 3. This calculation helps in recognizing the maximum uniformity among the numbers, which can be particularly useful in simplifying fractions or solving various arithmetic problems.

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Most popular questions from this chapter

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