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Chapter 6: Problem 2

Find the greatest common factor for each list of numbers. \(50,30,5\)

Short Answer

Expert verified
The greatest common factor is 5.

Step by step solution

01

- Find Prime Factors

Begin by finding the prime factors for each number. The prime factors of 50 are: 2, 5, 5 The prime factors of 30 are: 2, 3, 5 The prime factors of 5 are: 5
02

- Identify Common Factors

Identify the common prime factors in all the lists. Here, the only common prime factor in all three lists is 5.
03

- Determine the Greatest Common Factor

The greatest common factor (GCF) is the highest number that appears as a common factor. In this case, it is 5.

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Key Concepts

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

Prime Factors
To find the greatest common factor (GCF), we first need to understand prime factors. A prime factor is a factor that is a prime number. A prime number is a number greater than 1 that can only be divided by 1 and itself.
For example:
  • The prime factors of 50 are 2 and 5. When you break down 50, you get 2 脳 5 脳 5.
  • The prime factors of 30 are 2, 3, and 5. Breaking down 30 gives 2 脳 3 脳 5.
  • The prime factor of 5 is just 5, because 5 is already a prime number.
Finding the prime factors is an important first step in calculating the GCF.
Common Factors
After identifying the prime factors for each number, the next step is to find common factors. Common factors are numbers that appear in all the lists of prime factors.
In our example:
  • The prime factors of 50 include 2 and 5.
  • The prime factors of 30 include 2, 3, and 5.
  • The prime factor of 5 is just 5.
Here, the only common prime factor in all three lists is 5.
Identifying common factors allows us to move to the last step.
GCF Calculation
The final step is to determine the Greatest Common Factor (GCF). The GCF is the highest number that appears as a common factor.
In our example, the common factor from all three numbers is 5.
So, the GCF of 50, 30, and 5 is 5.
Using prime factors and identifying common factors makes finding the GCF straightforward and clear.

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Most popular questions from this chapter

Factor completely. If the polynomial cannot be factored, write prime. $$ -39+10 x+x^{2} $$

Solve each equation, and check your solutions. $$ 6 p^{2}(p+1)=4(p+1)-5 p(p+1) $$

Factor each polynomial. $$ (x+y) n^{2}+(x+y) n-20(x+y) $$

Factor completely. $$ y^{3} z+3 y^{2} z^{2}-54 y z^{3} $$

If a trinomial has a negative coefficient for the squared term, as in \(-2 x^{2}+11 x-12,\) it is usually easier to factor by first factoring out the common factor \(-1 .\) $$ \begin{aligned} -2 x^{2}+11 x-12 \\ =&-1\left(2 x^{2}-11 x+12\right) \\ =&-1(2 x-3)(x-4) \end{aligned} $$ Use this method to factor each trinomial. See Example 7(b). $$ $$ -2 a^{2}-5 a b-2 b^{2} $$
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