2.4 Find Multiples and Factors
3 min read•june 24, 2024
and are the building blocks of number relationships. They help us understand how numbers connect and divide. Knowing these patterns makes it easier to do math in our heads and spot connections between different numbers.
Prime and composite numbers are key players in the number world. Primes are like the atoms of math, while composites are made up of these basic units. This knowledge is super useful for breaking down big numbers and solving tricky math problems.
Understanding Multiples and Factors
Patterns of number multiples
Top images from around the web for Patterns of number multiples
Category:Matrix - Wikimedia Commons View original
Is this image relevant?
Category:Multiples - Wikimedia Commons View original
Is this image relevant?
Use the Language of Algebra · Intermediate Algebra View original
Is this image relevant?
Category:Matrix - Wikimedia Commons View original
Is this image relevant?
Category:Multiples - Wikimedia Commons View original
Is this image relevant?
1 of 3
Top images from around the web for Patterns of number multiples
Category:Matrix - Wikimedia Commons View original
Is this image relevant?
Category:Multiples - Wikimedia Commons View original
Is this image relevant?
Use the Language of Algebra · Intermediate Algebra View original
Is this image relevant?
Category:Matrix - Wikimedia Commons View original
Is this image relevant?
Category:Multiples - Wikimedia Commons View original
Is this image relevant?
1 of 3
- Multiples are the products obtained by multiplying a number by any integer (positive or negative)
- Multiples of 3 include 3, 6, 9, 12, 15, 18, ... and -3, -6, -9, -12, -15, -18, ...
- Even number multiples always result in even products
- Odd number multiples alternate between odd and even products
- Multiples of 7: 7 (odd), 14 (even), 21 (odd), 28 (even), ...
- Multiples of 5 consistently end with either 0 or 5 (0, 5, 10, 15, 20, 25, ...)
- Multiples of 10 always terminate with 0 (10, 20, 30, 40, ...)
Divisibility rules for 2, 3, 5, 10
- A number is divisible by 2 if its ones place digit is even (0, 2, 4, 6, or 8)
- 234 is divisible by 2 because 4 is even
- A number is divisible by 3 if the sum of its digits can be divided by 3
- 135 is divisible by 3 since 1 + 3 + 5 = 9, which is divisible by 3
- A number is divisible by 5 if it ends with either 0 or 5
- 275 and 480 are divisible by 5
- A number is divisible by 10 if it ends with 0
- 350 and 1,240 are divisible by 10
- are part of and help determine factors quickly
Factors and Number Classifications
Systematic factor identification
- Factors are numbers that divide evenly into another number, leaving no remainder
- Factors of 18 include 1, 2, 3, 6, 9, and 18
- Systematic methods to find all factors of a number:
- List all from 1 to the given number and check for
- Identify by dividing the number by integers from 1 up to its square root
- For 36, check integers from 1 to
- Factor pairs: 1 and 36 (1 × 36 = 36), 2 and 18 (2 × 18 = 36), 3 and 12 (3 × 12 = 36), 4 and 9 (4 × 9 = 36), 6 and 6 (6 × 6 = 36)
Prime vs composite numbers
- Prime numbers have exactly two factors: 1 and the number itself
- Examples of prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, ...
- Composite numbers have more than two factors
- Examples of composite numbers: 4, 6, 8, 9, 10, 12, 14, 15, 16, ...
- The number 1 is unique, as it is neither prime nor composite (1 has only one factor, itself)
Number Theory Concepts
- Number theory is the study of integers and their properties
- Integers are whole numbers, including positive and negative numbers, and zero
- Factor pairs are two numbers that, when multiplied together, produce a given number
- Understanding these concepts helps in analyzing the of numbers and their relationships
Key Terms to Review (20)
∈: The symbol '∈' is used to denote set membership, indicating that an element belongs to a specific set. It is a fundamental concept in set theory and is crucial for understanding the topics of finding multiples and factors.
÷: The division symbol, also known as the obelus, represents the mathematical operation of division. It indicates that the number or expression to the left of the symbol is to be divided by the number or expression to the right of the symbol. This key term is crucial in understanding various mathematical concepts, including finding multiples and factors, solving equations using integers, and multiplying and dividing fractions and mixed numbers.
Composite Number: A composite number is a positive integer that has at least one positive divisor other than 1 or the number itself. In other words, a composite number can be expressed as a product of two or more smaller integers.
Divisibility: Divisibility is the property of a number being evenly divisible by another number without a remainder. It is a fundamental concept in mathematics that underpins various topics, including dividing whole numbers, finding multiples and factors, and prime factorization.
Divisibility Rules: Divisibility rules are a set of guidelines that help determine whether a given whole number is divisible by another whole number without leaving a remainder. These rules are particularly useful in the context of dividing whole numbers, finding multiples and factors, and performing arithmetic operations with fractions.
Division: Division is a fundamental mathematical operation that involves partitioning a quantity into equal parts or groups. It represents the inverse of multiplication, allowing us to find how many times one number is contained within another. This key term is essential in understanding various mathematical concepts, from whole numbers to exponents and scientific notation.
Division Method: The division method is a mathematical operation that involves dividing one number by another to find the quotient. It is a fundamental operation in arithmetic and algebra, and is used to solve a variety of problems, including finding multiples and factors of numbers.
Even Division: Even division refers to the process of dividing a number by another number such that the result is a whole number without any remainder. It is a fundamental concept in mathematics that is closely tied to the ideas of factors and multiples.
Factor Pairs: Factor pairs refer to the combinations of two numbers that, when multiplied together, result in a given product. They represent the different ways a number can be expressed as the product of two factors.
Factor Tree: A factor tree is a visual representation used to find the prime factorization of a number. It involves breaking down a composite number into its prime factors in a hierarchical, tree-like structure.
Factors: Factors are whole numbers that can be multiplied together to produce another number. They play a critical role in various mathematical operations and help in understanding the relationships between numbers. Recognizing factors is essential for simplifying expressions, performing operations, and understanding properties of numbers.
Greatest Common Factor: The greatest common factor (GCF) is the largest positive integer that divides each of the given integers without a remainder. It is an essential concept in number theory and is closely related to finding multiples, prime factorization, and factoring polynomials.
Integers: Integers are a set of positive and negative whole numbers, including zero. They are the foundation for many mathematical operations and concepts, and are essential in understanding and working with various topics in pre-algebra.
Least Common Multiple: The least common multiple (LCM) is the smallest positive integer that is divisible by two or more given integers. It is a fundamental concept in mathematics that is closely related to finding factors and multiples of numbers, as well as understanding the relationships between fractions and decimals.
Multiples: Multiples are the products of a given number and any whole number. They represent the numbers in the multiplication table for that particular number, and are essential in understanding concepts like factors and divisibility.
Multiplication: Multiplication is a mathematical operation that involves the repeated addition of a number to itself. It is one of the four basic arithmetic operations, along with addition, subtraction, and division. Multiplication is used to find the total number of items or the area of a rectangle, and it is a fundamental concept in various mathematical contexts, including algebra, geometry, and statistics.
Multiplication Symbol (×): The multiplication symbol, denoted as '×', is a mathematical operation that represents the repeated addition of a number to itself. It is a fundamental operation in arithmetic and is used to find the product of two or more numbers. The multiplication symbol is a crucial component in various mathematical contexts, including whole numbers, fractions, and mixed numbers.
Number Theory: Number theory is the study of the properties and relationships of integers, or whole numbers. It explores fundamental questions about the nature of numbers and their behavior, focusing on concepts like divisibility, prime numbers, and patterns within the number system.
Prime Factorization: Prime factorization is the process of expressing a number as a product of its prime factors. It involves breaking down a number into the smallest possible prime numbers that, when multiplied together, result in the original number. This concept is fundamental in understanding various mathematical operations and relationships, including finding multiples and factors, as well as adding and subtracting fractions with different denominators.
Prime Number: A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself. Prime numbers are fundamental building blocks of the number system and play a crucial role in various mathematical concepts, including finding multiples and factors.