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Chapter 4: Problem 48

Find the GCF of each set of numbers or monomials. $$25 a, 30 b$$

Short Answer

Expert verified
The GCF of 25a and 30b is 5.

Step by step solution

01

Finding Prime Factors of Numbers

First, write down the prime factors of each number. The prime factors of 25 are: \(25 = 5 \times 5\). The prime factors of 30 are: \(30 = 2 \times 3 \times 5\).
02

Identifying Common Prime Factors

Observe the prime factors of both numbers. The common prime factor between 25 and 30 is \(5\).
03

Check for Common Variable Factors

Analyze the monomials to observe that there are no common variable factors between \(a\) and \(b\) as they are different.
04

Multiply the Common Factors

Only the number \(5\) is common. Thus, the greatest common factor (GCF) of the numbers is \(5\). No variables are common so the GCF of the entire expression 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 of breaking down a number into its basic building blocks, which are prime numbers. These are numbers greater than 1 that have no other divisors except 1 and themselves.
To find the prime factors of a number, you repeatedly divide the number by the smallest possible prime number until you're left with 1.
  • Start with the smallest prime number鈥攚hich is 2鈥攁nd check if the number is divisible by it.
  • If yes, divide and proceed with the quotient.
  • If not, try the next smallest prime, which is 3, and continue the process.
Each of these primes multiplied together equals the original number. By expressing numbers as products of their prime factors, it's simpler to identify common factors, like in the case of finding the GCF.
Common Factors
Common factors are numbers that are divisors of two or more numbers. To identify common factors, you take the prime factorization of each number and compare them.
Identify which prime numbers appear in both factorizations.
  • For instance, if 25 is broken down into its prime factors as \(5 \times 5\), and 30 into \(2 \times 3 \times 5\), you'll see the number 5 is present in both.
  • The common prime factors become the building blocks for calculating the GCF.
The common factors usually multiply together to form the GCF, which is the largest number that can evenly divide all of the included numbers or expressions.
Monomials
Monomials are algebraic expressions that consist of a single term. Each term is a product of a number and one or more variables raised to whole number powers.
For example, in the original set of monomials \(25a\) and \(30b\), each is considered separate because they contain different variables, \(a\) and \(b\).
  • In mathematical calculations involving monomials, treating the numerical coefficients and variable parts separately helps simplify the problem.
  • For example, the GCF of these monomials only involves the number part since the variables differ.
Handling the numeric and variable parts distinctly is crucial for correctly determining common factors and thus finding solutions to expressions with more than one term.

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Most popular questions from this chapter

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. $$\frac{9}{15}$$

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