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Finding the Greatest Common Factor
Factoring Trinomials
Absolute Value Function
A Summary of Factoring Polynomials
Solving Equations with One Radical Term
Adding Fractions
Subtracting Fractions
The FOIL Method
Graphing Compound Inequalities
Solving Absolute Value Inequalities
Adding and Subtracting Polynomials
Using Slope
Solving Quadratic Equations
Multiplication Properties of Exponents
Completing the Square
Solving Systems of Equations by using the Substitution Method
Combining Like Radical Terms
Elimination Using Multiplication
Solving Equations
Pythagoras' Theorem 1
Finding the Least Common Multiples
Multiplying and Dividing in Scientific Notation
Adding and Subtracting Fractions
Solving Quadratic Equations
Adding and Subtracting Fractions
Multiplication by 111
Adding Fractions
Multiplying and Dividing Rational Numbers
Multiplication by 50
Solving Linear Inequalities in One Variable
Simplifying Cube Roots That Contain Integers
Graphing Compound Inequalities
Simple Trinomials as Products of Binomials
Writing Linear Equations in Slope-Intercept Form
Solving Linear Equations
Lines and Equations
The Intercepts of a Parabola
Absolute Value Function
Solving Equations
Solving Compound Linear Inequalities
Complex Numbers
Factoring the Difference of Two Squares
Multiplying and Dividing Rational Expressions
Adding and Subtracting Radicals
Multiplying and Dividing Signed Numbers
Solving Systems of Equations
Factoring Out the Opposite of the GCF
Multiplying Special Polynomials
Properties of Exponents
Scientific Notation
Multiplying Rational Expressions
Adding and Subtracting Rational Expressions With Unlike Denominators
Multiplication by 25
Decimals to Fractions
Solving Quadratic Equations by Completing the Square
Quotient Rule for Exponents
Simplifying Square Roots
Multiplying and Dividing Rational Expressions
Independent, Inconsistent, and Dependent Systems of Equations
Graphing Lines in the Coordinate Plane
Graphing Functions
Powers of Ten
Zero Power Property of Exponents
The Vertex of a Parabola
Rationalizing the Denominator
Test for Factorability for Quadratic Trinomials
Trinomial Squares
Solving Two-Step Equations
Solving Linear Equations Containing Fractions
Multiplying by 125
Exponent Properties
Multiplying Fractions
Adding and Subtracting Rational Expressions With the Same Denominator
Quadratic Expressions - Completing Squares
Adding and Subtracting Mixed Numbers with Different Denominators
Solving a Formula for a Given Variable
Factoring Trinomials
Multiplying and Dividing Fractions
Multiplying and Dividing Complex Numbers in Polar Form
Power Equations and their Graphs
Solving Linear Systems of Equations by Substitution
Solving Polynomial Equations by Factoring
Laws of Exponents
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Systems of Linear Equations
Properties of Rational Exponents
Power of a Product and Power of a Quotient
Factoring Differences of Perfect Squares
Dividing Fractions
Factoring a Polynomial by Finding the GCF
Graphing Linear Equations
Steps in Factoring
Multiplication Property of Exponents
Solving Systems of Linear Equations in Three Variables
Solving Exponential Equations
Finding the GCF of a Set of Monomials
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Finding the Least Common Multiples

The multiples of a number are the products of that number and the whole numbers. For instance the multiples of 5 are the following.

A number that is a multiple of two or more numbers is called a common multiple of these numbers. To find the common multiples of 6 and 8, we first list the multiples of 6 and the multiples of 8 separately.

So the common multiples of 6 and 8 are 0, 24, 48, ... . Of the nonzero common multiples, the least common multiple of 6 and 8 is 24.


The least common multiple (LCM) of two or more numbers is the smallest nonzero number that is a multiple of each number.

A shortcut for finding the LCM—faster than listing multiples—involves prime factorization.

To Compute the Least Common Multiple (LCM)

  • find the prime factorization of each number,
  • identify the prime factors that appear in each factorization, and
  • multiply these prime factors, using each factor the greatest number of times that itoccurs in any of the factorizations.


Find the LCM of 8 and 12.


We first find the prime factorization of each number.

The factor 2 appears three times in the factorization of 8 and twice in the factorization of 12, so it must be included three times in forming the least common multiple.

As always, it is a good idea to check that our answer makes sense. We do so by verifying that 8 and 12 really are factors of 24.


Find the LCM of 5 and 9.


First we write each number as the product of primes.

To find the LCM we multiply the highest power of each prime.

So the LCM of 5 and 9 is 45. Note that 45 is also the product of 5 and 9. Checking our answer, we see that 45 is a multiple of both 5 and 9.


If two or more numbers have no common factor (other than 1), the LCM is their product.

Now let's find the LCM of three numbers.


Find the LCM of 3, 5, and 6.


First we find the prime factorizations of these three numbers.

3 = 3

5 = 5

6 = 2 × 3

The LCM is therefore the product 2 × 3 × 5, which is 30. Note that 30 is a multiple of 3, 5, and 6, which supports our answer.


A gym that is open every day of the week offers aerobic classes everythird day and gymnastic classes every fourth day. You took both classesthis morning. In how many days will the gym offer both classes on thesame day?


To answer this question, we ask: What is the LCM of 3 and 4? As usual, we begin by finding prime factorizations.

To find the LCM, we multiply 3 by .

Both classes will be offered again on the same day in 12 days.

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