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Chapter 5: Problem 8

Find the greatest common factor for each list of numbers. 15,30,45,75

Short Answer

Expert verified
The greatest common factor is 15.

Step by step solution

01

List the factors of each number

First, list out all the factors for each of the given numbers. Factors of 15: 1, 3, 5, 15 Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30 Factors of 45: 1, 3, 5, 9, 15, 45 Factors of 75: 1, 3, 5, 15, 25, 75
02

Identify the common factors

Next, find the common factors that are present in each list of factors. Common factors of 15, 30, 45, and 75: 1, 3, 5, 15
03

Determine the greatest common factor

Finally, out of the common factors, identify the greatest one. The greatest common factor (GCF) is 15.

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Key Concepts

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

Factors
In mathematics, a factor of a number is a number that divides into it with no remainder. For example, let's take the number 15. We can find its factors by identifying all the numbers that can divide 15 completely:
  • 1 (since 15 梅 1 = 15)
  • 3 (since 15 梅 3 = 5)
  • 5 (since 15 梅 5 = 3)
  • 15 (since 15 梅 15 = 1)
This means the factors of 15 are 1, 3, 5, and 15.
Factors can be visualized as the building blocks of a number. Knowing them helps in many mathematical operations such as simplifying fractions and finding the greatest common factor (GCF).
Common Factors
Common factors are the numbers that are factors of two or more numbers. They are the values that two or more numbers share in their factor lists. For instance, let's find the common factors of 15 and 30 by first listing out their factors:
  • Factors of 15: 1, 3, 5, 15
  • Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
To identify the common factors of these numbers, we look for factors that appear in both lists:
  • Common factors of 15 and 30: 1, 3, 5, 15
Common factors are useful in simplifying fractions and finding the greatest common factor (GCF). They highlight the numbers' shared properties, making them easier to work with in equations and problems.
GCF Calculation
The greatest common factor (GCF) is the largest number that is a common factor of two or more numbers. To find the GCF of a list of numbers, we follow a systematic process:
  • Step 1: List the factors of each number. For example:
    • Factors of 15: 1, 3, 5, 15
    • Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30
    • Factors of 45: 1, 3, 5, 9, 15, 45
    • Factors of 75: 1, 3, 5, 15, 25, 75

  • Step 2: Identify the common factors. In this case, the common factors of 15, 30, 45, and 75 are: 1, 3, 5, 15.

  • Step 3: Determine the greatest common factor. From the common factors, the largest number is 15.
    This makes the GCF of 15, 30, 45, and 75 equal to 15.
Finding the GCF is crucial in mathematical operations like simplifying fractions, solving rational expressions, and working through algebraic problems. It allows you to reduce complex numbers to their simplest form and makes calculations more manageable.

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Most popular questions from this chapter

Solve each problem. Find three consecutive odd integers such that 3 times the sum of all three is 18 more than the product of the first and second integers.

Find the greatest common factor for each list of terms. \(45 c^{3} d, 75 c, 90 d, 105 c d\)

Factor each trinomial completely. \(2 t^{2}-14 t+15\)

On a quiz, a student factored \(16 x^{2}-24 x+5\) by grouping as follows, but he did not receive credit for his answer. \(\begin{aligned} 16 x^{2}-24 x+5 \\\ &=16 x^{2}-4 x-20 x+5 \\ &=4 x(4 x-1)-5(4 x-1) \end{aligned}\) Give the correct factored form.

Factor each polynomial completely. \(9 a^{2}-48 a b+64 b^{2}-4 c^{2}\)
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