Factors of a number are the integers that can be multiplied together to produce that number. In simpler terms, if you can divide one number by another without leaving a remainder, then the second number is a factor of the first. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
Example: Finding Factors
To find the factors of 24, you start with 1 and 24, then proceed to check each integer up to the square root of 24:
Step 1: 1 × 24 = 24
Step 2: 2 × 12 = 24
Step 3: 3 × 8 = 24
Step 4: 4 × 6 = 24
Answer: The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
Multiples of a number are the products obtained by multiplying that number by any integer. Unlike factors, multiples can be infinitely many. For example, the multiples of 5 include 5, 10, 15, 20, and so on.
Example: Finding Multiples
To find the first five multiples of 7, simply multiply 7 by the integers 1 through 5:
Step 1: 7 × 1 = 7
Step 2: 7 × 2 = 14
Step 3: 7 × 3 = 21
Step 4: 7 × 4 = 28
Step 5: 7 × 5 = 35
Answer: The first five multiples of 7 are 7, 14, 21, 28, and 35.
Prime factorization is the process of breaking down a number into its prime factors, which are factors that are prime numbers. This is a critical concept in pre-algebra and is used in various applications, including finding the greatest common divisor (GCD) and the least common multiple (LCM).
Example: Prime Factorization
To find the prime factorization of 60:
Step 1: Divide by the smallest prime number, 2: 60 ÷ 2 = 30
Step 2: Divide 30 by 2: 30 ÷ 2 = 15
Step 3: Divide 15 by the next smallest prime, 3: 15 ÷ 3 = 5
Answer: The prime factorization of 60 is 2 × 2 × 3 × 5 or 22 × 3 × 5.
Common factors are factors that two or more numbers have in common. The greatest common factor (GCF) is the largest of these. Common multiples are multiples that are shared by two or more numbers, and the least common multiple (LCM) is the smallest of these.
Example: Finding the GCF
To find the GCF of 18 and 24:
Step 1: List the factors of 18: 1, 2, 3, 6, 9, 18
Step 2: List the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Answer: The common factors are 1, 2, 3, 6, and the GCF is 6.
Example: Finding the LCM
To find the LCM of 4 and 6:
Step 1: List the multiples of 4: 4, 8, 12, 16, 20...
Step 2: List the multiples of 6: 6, 12, 18, 24...
Answer: The common multiples are 12, 24, and the LCM is 12.
Factors and multiples have real-world applications in various fields, including mathematics, science, and engineering. For example, they are used in simplifying fractions, solving equations, and analyzing periodic patterns.
Simplifying fractions involves dividing the numerator and the denominator by their GCF. This makes the fraction easier to work with in calculations.
Example: Simplifying Fractions
Simplify the fraction 18/24:
Step 1: Find the GCF of 18 and 24, which is 6.
Step 2: Divide both the numerator and denominator by 6: (18 ÷ 6) / (24 ÷ 6) = 3/4.
Answer: The simplified fraction is 3/4.
Factors and multiples are often used in solving equations, particularly those that involve polynomial expressions. Understanding the factors of an equation can help in finding its roots and simplifying the problem.
In science and engineering, factors and multiples are used to analyze periodic patterns, such as sound waves, electrical signals, and other cyclical phenomena. Recognizing the factors and multiples of a pattern can reveal underlying structures and help in predicting future behavior.
