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Chapter 5: Problem 6

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

Short Answer

Expert verified
The greatest common factor for 50, 30, and 5 is 5.

Step by step solution

01

List the factors of each number

List all the factors for each given number.Factors of 50: 1, 2, 5, 10, 25, 50Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30Factors of 5: 1, 5
02

Identify common factors

From the lists of factors, identify the common factors. The common factors for 50, 30, and 5 are: 1 and 5.
03

Find the greatest common factor

Determine the greatest factor that is common to all lists. Here, the greatest common factor is 5.

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Key Concepts

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

factorization
Factorization is the process of breaking down a number into its smallest divisible components. These components are called factors.
For instance, let's take the number 50. When we perform factorization on 50, we divide it by all possible integers that result in whole numbers:
  • 50 梅 1 = 50
  • 50 梅 2 = 25
  • 50 梅 5 = 10
  • 50 梅 10 = 5
  • 50 梅 25 = 2
  • 50 梅 50 = 1
As a result, the factors of 50 are 1, 2, 5, 10, 25, and 50.
This process helps us break down any number into its basic components, allowing us to find commonality among multiple numbers.
common factors
Common factors are the factors that are shared between two or more numbers.
To determine common factors, we first need to list the factors for each number.
For example, let's consider 30 and 50.
  • Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
  • Factors of 50: 1, 2, 5, 10, 25, 50
Now, we can see that the numbers 1, 2, 5, and 10 are present in both lists. These are the common factors.
Identifying these shared factors is crucial for finding the greatest common factor, as only these shared factors are considered in the next steps.
greatest common divisor
The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest number that is a factor of two or more numbers.
After identifying common factors, we simply pick the highest one.
Continuing our previous example with the numbers 30 and 50, we found the common factors to be 1, 2, 5, and 10.
Among these, 10 is the greatest.
  • If we add number 5 to our list and find common factors of 30, 50, and 5, the common factors are now 1 and 5.
  • Thus, the GCD of 30, 50, and 5 is 5.
Finding the GCD is extremely useful in simplifying fractions, solving problems in number theory, and more.

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Most popular questions from this chapter

Factor each trinomial. \(36 x^{4}-64 x^{2} y+15 y^{2}\)

Factor each trinomial completely. \(12 k^{3} q^{4}-4 k^{2} q^{5}-k q^{6}\)

Solve each problem. A lot has the shape of a right triangle with one leg \(2 \mathrm{~m}\) longer than the other. The hypotenuse is \(2 \mathrm{~m}\) less than twice the length of the shorter leg. Find the length of the shorter leg.

Factor each binomial completely. \(y^{3}-216\)

Solve each problem. The product of the first and third of three consecutive integers is 3 more than 3 times the second integer. Find the integers.
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