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Chapter 6: Problem 66

Find the greatest common factor of each set of numbers. $$ 15,30,65 $$

Short Answer

Expert verified
The greatest common factor is 5.

Step by step solution

01

Prime Factorization

First, find the prime factorization of each number. - 15: The prime factorization is \(3 \times 5\).- 30: The prime factorization is \(2 \times 3 \times 5\).- 65: The prime factorization is \(5 \times 13\).
02

Identify Common Prime Factors

Identify the prime factors that are common to all three numbers. The only common prime factor among 15, 30, and 65 is \(5\).
03

Greatest Common Factor

The greatest common factor (GCF) is the product of all common prime factors identified in Step 2. Since \(5\) is the only common factor, the GCF is \(5\).

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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 method used to express a number as a product of its prime factors. To perform prime factorization, you break down the number into the smallest parts possible鈥攑rime numbers. These are numbers that have only two divisors: 1 and themselves. For example, 2, 3, 5, 7, and 11 are all prime numbers.
To find the prime factorization of a number, you divide the number by the smallest prime number until it cannot be divided further. For instance:
  • The number 15 can be divided by 3, resulting in 5. Since 5 is a prime number, the prime factorization of 15 is \(3 \times 5\).
  • The number 30 is divisible by 2, giving 15, which can then be divided by 3 to give 5. Thus, the prime factorization of 30 is \(2 \times 3 \times 5\).
  • For 65, dividing by 5 gives 13. Since 13 is prime, the prime factorization of 65 is \(5 \times 13\).
Common Prime Factors
Common prime factors are those prime numbers that appear in the prime factorizations of two or more numbers. To identify them, compare the prime factors of each number and note those that appear across all lists. This step is crucial because it determines what the numbers share in terms of their building blocks.
In the given problem:
  • The prime factorization of 15 is \(3 \times 5\),
  • The prime factorization of 30 is \(2 \times 3 \times 5\),
  • And the prime factorization of 65 is \(5 \times 13\).
Here, the number 5 is highlighted as the only prime factor common to all three numbers: 15, 30, and 65.
Greatest Common Divisor (GCD)
The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest number that divides two or more numbers without a remainder. Finding the GCD helps simplify fractions and solve problems involving ratios.

To determine the GCD, follow these steps:
  • Make sure to perform prime factorization for each number in the list.
  • Identify the common prime factors that you found previously.
  • Multiply all the common prime factors together. If there's more than one common factor, multiply them together; if a common factor repeats, include its power.
In our example, the common prime factor among 15, 30, and 65 was 5. Since we only have one common factor, our GCD is simply 5. Thus, the greatest common factor of 15, 30, and 65 is 5.

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Most popular questions from this chapter

Use a graphing calculator to estimate the \(x\) -coordinates at which the maxima and minima of each function occur. Round to the nearest hundredth. $$ f(x)=-x^{3}+6 x^{2}-6 x-5 $$

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. $$ x^{3}+2 x^{2}-x-2 ; x-1 $$

Factor completely. If the polynomial is not factorable, write prime. $$ 7 x y^{3}-14 x^{2} y^{5}+28 x^{3} y^{2} $$

Graph each function by making a table of values. $$ f(x)=x^{3}-4 x^{2}+x+5 $$

Use synthetic substitution to find \(g(3)\) and \(g(-4)\) for each function. $$ g(x)=x^{3}+2 x^{2}-3 x+1 $$
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