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Chapter 5: Problem 11

Find the greatest common factor for each list of terms. \(16 y, 24\)

Short Answer

Expert verified
The GCF of 16 and 24 is 8.

Step by step solution

01

- Factor each number

Factorize each number separately. Identify the prime factors for each term.For 16: 16 can be factored into prime factors as follows:16 = 2 x 2 x 2 x 2 = 2^4For 24:24 can be factored into prime factors as follows:24 = 2 x 2 x 2 x 3 = 2^3 x 3
02

- Identify common factors

Compare the factorizations of each number and identify the common factors.For 16: 2^4For 24: 2^3 x 3The common prime factor is 2. The smallest power of this common factor is 2^3.
03

- Multiply common factors

Multiply the common factors using the smallest power identified.Since the smallest power of 2 in both factorizations is 2^3, the greatest common factor is calculated as: 2^3 = 8

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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 a method used to break down a number into its smallest prime factors. Prime numbers are numbers greater than 1 with only two factors: 1 and the number itself. To factorize a number, repeatedly divide it by its smallest prime factors.
For example, let's factorize 16 and 24:
  • 16 can be factorized as: 16 = 2 脳 2 脳 2 脳 2 = 2^4
  • 24 can be factorized as: 24 = 2 脳 2 脳 2 脳 3 = 2^3 脳 3
Breaking numbers down into their prime components makes it easier to identify common factors and perform other calculations.
common factors
Common factors are factors that two or more numbers share. After performing the prime factorization, we compare the prime factors of each number to identify any common factors.
For the numbers in the example (16 and 24), their prime factorization is:
  • 16: 2^4
  • 24: 2^3 脳 3
The common prime factor here is 2. It's present in both 16 and 24.
To find the greatest common factor, we take the lowest power of all the common prime factors. So, the lowest power of 2 in this case is 2^3.
math steps
Let's break down the steps to find the Greatest Common Factor (GCF) of two numbers:
1. **Factor each number**: Factorize both numbers separately into their prime factors.
  • For 16: 16 = 2^4
  • For 24: 24 = 2^3 脳 3
2. **Identify common factors**: Compare the prime factorizations and find the common prime factors.
For 16 and 24, the common factor is 2.
3. **Multiply common factors**: Use the smallest power of the common factors to calculate the GCF. For 2, the smallest power is 2^3.
So, the GCF of 16 and 24 is 2^3 = 8.
Following these **math steps** will help you find the greatest common factor easily.

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Most popular questions from this chapter

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Apply the special factoring rules of this section to factor each polynomial. \(q^{2}-\frac{1}{4}\)

The following powers of \(x\) are all perfect cubes: $$ x^{3}, x^{6}, \quad x^{9}, \quad x^{12}, \quad x^{15} $$ On the basis of this observation, we may make a conjecture that if the power of a variable is divisible by ____ (with 0 remainder), then we have a perfect cube.

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