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Chapter 5: Problem 5

Find the greatest common factor for each list of numbers. 40,20,4

Short Answer

Expert verified
The greatest common factor is 4.

Step by step solution

01

Prime Factorization of Each Number

Break down each number into its prime factors.\[40 = 2^3 \times 5\]\[20 = 2^2 \times 5\]\[4 = 2^2\]
02

Identify Common Prime Factors

Find the prime factors that are common to all the numbers. Here, the common prime factor is 2.
03

Determine the Lowest Power of Common Factors

Identify the lowest power of the common prime factor from each prime factorization. The lowest power of 2 is \(2^2\).
04

Calculate the Greatest Common Factor (GCF)

Multiply the lowest power of the common factors to get the GCF.\[GCF = 2^2 = 4\]

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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 composite number into prime numbers. A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. For instance, the number 40 can be broken down into its prime factors: 1. Start by dividing 40 by the smallest prime number, 2. 2. Continue dividing by 2 until you can't anymore: 40 梅 2 = 20, 20 梅 2 = 10, 10 梅 2 = 5. Now, 5 is a prime number, so you stop. This gives us: \[40 = 2^3 \times 5 \20 = 2^2 \times 5 \4 = 2^2\] You repeat this process for each of your numbers.
Common Prime Factors
Once you've done the prime factorization, identify any prime factors that each number shares. This is crucial for finding the greatest common factor (GCF). From our example: \[40 = 2^3 \times 5 \x \20 = 2^2 \times 5 \x \4 = 2^2\] Here, the common prime factor between 40, 20, and 4 is 2. Finding these common primes helps us take the next step towards determining the GCF.
Lowest Power of Common Factors
After identifying the common prime factors, the next step is to determine the lowest power of these common factors. From the example: \[40 = 2^3 \x \20 = 2^2 \x \4 = 2^2\] The common prime factor is 2. The lowest power of 2 in the factorizations is \[2^2\]. Lowest power is crucial because we want the largest possible factor that divides all given numbers without any remainder.
GCF Calculation
Now it's time to calculate the Greatest Common Factor (GCF). Using the lowest power of the common prime factors identified, we multiply them: For our example, the lowest power of 2 is \[2^2\]. So, \[GCF = 2^2 = 4\] This gives us the greatest common factor for the numbers 40, 20, and 4, which is 4. Remember, the GCF is the highest number that can divide all the given numbers.

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Most popular questions from this chapter

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