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Chapter 2: Problem 68

Find the GCF. $$14,42,84$$

Short Answer

Expert verified
The GCF of 14, 42, and 84 is 14.

Step by step solution

01

Factorize

First, factorize the three numbers.\n14 can be factorized into \(2 \times 7\).\n42 can be factorized into \(2 \times 3 \times 7\).\n84 can be factorized into \(2^2 \times 3 \times 7\).
02

Find the Common Factors

Next, find the common factors from the prime factorization prepared on step 1. The numbers that appear in all three sets are 2 and 7.
03

Multiply the Common Factors

Multiply these common factors together to obtain the GCF. The GCF would be \(2 \times 7 = 14\).

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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 the process of breaking down a number into its basic building blocks, which are prime numbers. A prime number is a number greater than 1 that has no divisors other than 1 and itself. To find the prime factorization, you divide a number by prime numbers until all factors are prime. Let鈥檚 go through the prime factorization of each number in the exercise:
  • For 14: The smallest prime number that divides 14 is 2, resulting in 14 divided by 2 equals 7. Both 2 and 7 are prime numbers, so the prime factorization of 14 is \(2 \times 7\).
  • For 42: Again, start with the smallest prime, which is 2. Divide 42 by 2 to get 21. Next, 21 is divisible by 3, resulting in 7. All numbers - 2, 3, and 7 - are prime, so 42 = \(2 \times 3 \times 7\).
  • For 84: Start with 2, dividing 84 by 2 gives 42. Divide 42 by 2 again to get 21. Then, divide 21 by 3 to reach 7, with 7 being prime. The prime factorization of 84 is \(2^2 \times 3 \times 7\).
This breakdown helps us see what each number is made of, allowing us to find common elements among different numbers.
Common Factors
When we talk about finding common factors, we鈥檙e looking for numbers that appear as factors in multiple numbers. With prime factorization, this is simpler because we focus on prime factors shared by all numbers involved. For the numbers 14, 42, and 84:
  • 14 is represented by \(2 \times 7\).
  • 42 is represented by \(2 \times 3 \times 7\).
  • 84 is represented by \(2^2 \times 3 \times 7\).
As we compare these, we look for the smallest powers of primes common to all factorizations. The common factors are:
  • The number 2, which appears in 14 and 42 with the smallest power of 1.
  • The number 7, present in all factorizations.
Identifying these shared numbers is what helps us find the greatest common factor.
Multiplication of Factors
Once you identify the common factors, the final step in determining the GCF (Greatest Common Factor) is by multiplying these together.From our previous findings, we have:
  • The common factors between 14, 42, and 84 are 2 and 7.
We multiply these common prime factors to find the GCF:
  • Multiply 2 and 7: \(2 \times 7 = 14\).
So, the GCF of 14, 42, and 84 is 14. This means 14 is the largest number that can evenly divide all three original numbers, showing the significance of understanding and applying multiplication of common factors effectively.

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