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Chapter 5: Problem 8

Find the greatest common factor. $$30,45,60$$

Short Answer

Expert verified
The greatest common factor of the numbers 30, 45, and 60 is 15.

Step by step solution

01

Prime Factorization

Find the prime factorization of all three given numbers. The prime factorization of 30 is \(2 \times 3 \times 5\). The prime factorization of 45 is \(3^2 \times 5\). Lastly, the prime factorization of 60 is \(2^2 \times 3 \times 5\).
02

Identify the common prime factors

Now identify what prime numbers are common factors of all three numbers. Here, the common prime factors are 3 and 5.
03

Find the product of common prime factors

Take the minimum power of the common prime factors and multiply them. Both 3 and 5 appear once in every factorization at least. Therefore, the greatest common factor (GCF) is \(3 \times 5 = 15\) .

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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 key concept when learning about factors of numbers. This method involves breaking down a number into its simplest building blocks: prime numbers. To perform prime factorization, you divide the number by the smallest prime number possible and continue the process with the resulting quotient until all factors are prime.

For example:
  • 30 can be split as 2 脳 3 脳 5.
  • 45 breaks down into 3 脳 3 脳 5, which can also be written as \(3^2 \times 5\).
  • 60 can be factored into 2 脳 2 脳 3 脳 5, or \(2^2 \times 3 \times 5\).
Understanding prime factorization is critical, as it reveals the basic structure of a number and plays a crucial role in finding common factors.
Common Factors
Common factors are numbers that divide evenly into two or more numbers. They are vital when you want to simplify fractions, find greatest common factors, or solve problems involving ratios.

When prime factorizations of multiple numbers are available, identifying common factors becomes simple. It involves picking out all prime numbers that appear in each prime factorization. In the case of 30, 45, and 60:
  • The prime factors of 30 are 2, 3, and 5.
  • For 45, the prime factors are 3 and 5.
  • 60 has 2, 3, and 5 as prime factors.
The common factors are those that appear in all three sets. Here, the primes 3 and 5 are common factors. These common elements form the building blocks for determining the greatest common factor.
Multiplication of Factors
The greatest common factor (GCF) emerges from the multiplication of common prime factors. Once you've established which prime numbers are shared across all sets, you need to consider their smallest power present in each factorization.

In simpler terms, you look at how many times these common prime numbers appear in each list and use the lowest occurrences:
  • Both numbers 3 and 5 appear at least once in the factorizations of 30, 45, and 60.
  • So, you take \(3^1 \times 5^1 = 15\).
By multiplying these common factors, you find the greatest common factor, which represents the largest number that can evenly divide all the original numbers. Identifying and multiplying common factors is a fundamental step in problem-solving across mathematics.

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Most popular questions from this chapter

Two students factor \(2 x^{2}+20 x+42\) and get two different answers: \((2 x+6)(x+7)\) and \((x+3)(2 x+14)\) Do both answers check? Why don't they agree? Is either completely correct?

Factor. If the polynomial is prime, so indicate. $$24 a^{2}+14 a b+2 b^{2}$$

Think of two positive integers. Divide their product by their greatest common factor. Why do you think the result is called the least common multiple of the two integers? (Hint: The multiples of an integer such as 5 are \(5,10,15,20,25,30,\) and so on.

Factor. If the polynomial is prime, so indicate. $$30 r^{5}+63 r^{4}-30 r^{3}$$

For what values of \(b\) will the trinomial \(5 y^{2}-b y-3\) be factorable?
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