StudySmarter - The all-in-one study app.

4.8 • +11k Ratings

More than 3 Million Downloads

Free

Common Multiples

- Calculus
- Absolute Maxima and Minima
- Absolute and Conditional Convergence
- Accumulation Function
- Accumulation Problems
- Algebraic Functions
- Alternating Series
- Antiderivatives
- Application of Derivatives
- Approximating Areas
- Arc Length of a Curve
- Arithmetic Series
- Average Value of a Function
- Calculus of Parametric Curves
- Candidate Test
- Combining Differentiation Rules
- Combining Functions
- Continuity
- Continuity Over an Interval
- Convergence Tests
- Cost and Revenue
- Density and Center of Mass
- Derivative Functions
- Derivative of Exponential Function
- Derivative of Inverse Function
- Derivative of Logarithmic Functions
- Derivative of Trigonometric Functions
- Derivatives
- Derivatives and Continuity
- Derivatives and the Shape of a Graph
- Derivatives of Inverse Trigonometric Functions
- Derivatives of Polar Functions
- Derivatives of Sec, Csc and Cot
- Derivatives of Sin, Cos and Tan
- Determining Volumes by Slicing
- Direction Fields
- Disk Method
- Divergence Test
- Eliminating the Parameter
- Euler's Method
- Evaluating a Definite Integral
- Evaluation Theorem
- Exponential Functions
- Finding Limits
- Finding Limits of Specific Functions
- First Derivative Test
- Function Transformations
- General Solution of Differential Equation
- Geometric Series
- Growth Rate of Functions
- Higher-Order Derivatives
- Hydrostatic Pressure
- Hyperbolic Functions
- Implicit Differentiation Tangent Line
- Implicit Relations
- Improper Integrals
- Indefinite Integral
- Indeterminate Forms
- Initial Value Problem Differential Equations
- Integral Test
- Integrals of Exponential Functions
- Integrals of Motion
- Integrating Even and Odd Functions
- Integration Formula
- Integration Tables
- Integration Using Long Division
- Integration of Logarithmic Functions
- Integration using Inverse Trigonometric Functions
- Intermediate Value Theorem
- Inverse Trigonometric Functions
- Jump Discontinuity
- Lagrange Error Bound
- Limit Laws
- Limit of Vector Valued Function
- Limit of a Sequence
- Limits
- Limits at Infinity
- Limits of a Function
- Linear Approximations and Differentials
- Linear Differential Equation
- Linear Functions
- Logarithmic Differentiation
- Logarithmic Functions
- Logistic Differential Equation
- Maclaurin Series
- Manipulating Functions
- Maxima and Minima
- Maxima and Minima Problems
- Mean Value Theorem for Integrals
- Models for Population Growth
- Motion Along a Line
- Motion in Space
- Natural Logarithmic Function
- Net Change Theorem
- Newton's Method
- Nonhomogeneous Differential Equation
- One-Sided Limits
- Optimization Problems
- P Series
- Particle Model Motion
- Particular Solutions to Differential Equations
- Polar Coordinates
- Polar Coordinates Functions
- Polar Curves
- Population Change
- Power Series
- Ratio Test
- Removable Discontinuity
- Riemann Sum
- Rolle's Theorem
- Root Test
- Second Derivative Test
- Separable Equations
- Simpson's Rule
- Solid of Revolution
- Solutions to Differential Equations
- Surface Area of Revolution
- Symmetry of Functions
- Tangent Lines
- Taylor Polynomials
- Taylor Series
- Techniques of Integration
- The Fundamental Theorem of Calculus
- The Mean Value Theorem
- The Power Rule
- The Squeeze Theorem
- The Trapezoidal Rule
- Theorems of Continuity
- Trigonometric Substitution
- Vector Valued Function
- Vectors in Calculus
- Vectors in Space
- Washer Method
- Decision Maths
- Geometry
- 2 Dimensional Figures
- 3 Dimensional Vectors
- 3-Dimensional Figures
- Altitude
- Angles in Circles
- Arc Measures
- Area and Volume
- Area of Circles
- Area of Circular Sector
- Area of Parallelograms
- Area of Plane Figures
- Area of Rectangles
- Area of Regular Polygons
- Area of Rhombus
- Area of Trapezoid
- Area of a Kite
- Composition
- Congruence Transformations
- Congruent Triangles
- Convexity in Polygons
- Coordinate Systems
- Dilations
- Distance and Midpoints
- Equation of Circles
- Equilateral Triangles
- Figures
- Fundamentals of Geometry
- Geometric Inequalities
- Geometric Mean
- Geometric Probability
- Glide Reflections
- HL ASA and AAS
- Identity Map
- Inscribed Angles
- Isometry
- Isosceles Triangles
- Law of Cosines
- Law of Sines
- Linear Measure and Precision
- Median
- Parallel Lines Theorem
- Parallelograms
- Perpendicular Bisector
- Plane Geometry
- Polygons
- Projections
- Properties of Chords
- Proportionality Theorems
- Pythagoras Theorem
- Rectangle
- Reflection in Geometry
- Regular Polygon
- Rhombuses
- Right Triangles
- Rotations
- SSS and SAS
- Segment Length
- Similarity
- Similarity Transformations
- Special quadrilaterals
- Squares
- Surface Area of Cone
- Surface Area of Cylinder
- Surface Area of Prism
- Surface Area of Sphere
- Surface Area of a Solid
- Surface of Pyramids
- Symmetry
- Translations
- Trapezoids
- Triangle Inequalities
- Triangles
- Using Similar Polygons
- Vector Addition
- Vector Product
- Volume of Cone
- Volume of Cylinder
- Volume of Pyramid
- Volume of Solid
- Volume of Sphere
- Volume of prisms
- Mechanics Maths
- Acceleration and Time
- Acceleration and Velocity
- Angular Speed
- Assumptions
- Calculus Kinematics
- Coefficient of Friction
- Connected Particles
- Constant Acceleration
- Constant Acceleration Equations
- Converting Units
- Force as a Vector
- Kinematics
- Newton's First Law
- Newton's Law of Gravitation
- Newton's Second Law
- Newton's Third Law
- Projectiles
- Pulleys
- Resolving Forces
- Statics and Dynamics
- Tension in Strings
- Variable Acceleration
- Probability and Statistics
- Bar Graphs
- Basic Probability
- Charts and Diagrams
- Conditional Probabilities
- Continuous and Discrete Data
- Frequency, Frequency Tables and Levels of Measurement
- Independent Events Probability
- Line Graphs
- Mean Median and Mode
- Mutually Exclusive Probabilities
- Probability Rules
- Probability of Combined Events
- Quartiles and Interquartile Range
- Systematic Listing
- Pure Maths
- ASA Theorem
- Absolute Value Equations and Inequalities
- Addition and Subtraction of Rational Expressions
- Addition, Subtraction, Multiplication and Division
- Algebra
- Algebraic Fractions
- Algebraic Notation
- Algebraic Representation
- Analyzing Graphs of Polynomials
- Angle Measure
- Angles
- Angles in Polygons
- Approximation and Estimation
- Area and Circumference of a Circle
- Area and Perimeter of Quadrilaterals
- Area of Triangles
- Arithmetic Sequences
- Average Rate of Change
- Bijective Functions
- Binomial Expansion
- Binomial Theorem
- Chain Rule
- Circle Theorems
- Circles
- Circles Maths
- Combination of Functions
- Combinatorics
- Common Factors
- Common Multiples
- Completing the Square
- Completing the Squares
- Complex Numbers
- Composite Functions
- Composition of Functions
- Compound Interest
- Compound Units
- Conic Sections
- Construction and Loci
- Converting Metrics
- Convexity and Concavity
- Coordinate Geometry
- Coordinates in Four Quadrants
- Cubic Function Graph
- Cubic Polynomial Graphs
- Data transformations
- Deductive Reasoning
- Definite Integrals
- Deriving Equations
- Determinant of Inverse Matrix
- Determinants
- Differential Equations
- Differentiation
- Differentiation Rules
- Differentiation from First Principles
- Differentiation of Hyperbolic Functions
- Direct and Inverse proportions
- Disjoint and Overlapping Events
- Disproof by Counterexample
- Distance from a Point to a Line
- Divisibility Tests
- Double Angle and Half Angle Formulas
- Drawing Conclusions from Examples
- Ellipse
- Equation of Line in 3D
- Equation of a Perpendicular Bisector
- Equation of a circle
- Equations
- Equations and Identities
- Equations and Inequalities
- Estimation in Real Life
- Euclidean Algorithm
- Evaluating and Graphing Polynomials
- Even Functions
- Exponential Form of Complex Numbers
- Exponential Rules
- Exponentials and Logarithms
- Expression Math
- Expressions and Formulas
- Faces Edges and Vertices
- Factorials
- Factoring Polynomials
- Factoring Quadratic Equations
- Factorising expressions
- Factors
- Finding Maxima and Minima Using Derivatives
- Finding Rational Zeros
- Finding the Area
- Forms of Quadratic Functions
- Fractional Powers
- Fractional Ratio
- Fractions
- Fractions and Decimals
- Fractions and Factors
- Fractions in Expressions and Equations
- Fractions, Decimals and Percentages
- Function Basics
- Functional Analysis
- Functions
- Fundamental Counting Principle
- Fundamental Theorem of Algebra
- Generating Terms of a Sequence
- Geometric Sequence
- Gradient and Intercept
- Graphical Representation
- Graphing Rational Functions
- Graphing Trigonometric Functions
- Graphs
- Graphs and Differentiation
- Graphs of Common Functions
- Graphs of Exponents and Logarithms
- Graphs of Trigonometric Functions
- Greatest Common Divisor
- Growth and Decay
- Growth of Functions
- Highest Common Factor
- Hyperbolas
- Imaginary Unit and Polar Bijection
- Implicit differentiation
- Inductive Reasoning
- Inequalities Maths
- Infinite geometric series
- Injective functions
- Instantaneous Rate of Change
- Integers
- Integrating Polynomials
- Integrating Trig Functions
- Integrating e^x and 1/x
- Integration
- Integration Using Partial Fractions
- Integration by Parts
- Integration by Substitution
- Integration of Hyperbolic Functions
- Interest
- Inverse Hyperbolic Functions
- Inverse Matrices
- Inverse and Joint Variation
- Inverse functions
- Iterative Methods
- Law of Cosines in Algebra
- Law of Sines in Algebra
- Laws of Logs
- Limits of Accuracy
- Linear Expressions
- Linear Systems
- Linear Transformations of Matrices
- Location of Roots
- Logarithm Base
- Logic
- Lower and Upper Bounds
- Lowest Common Denominator
- Lowest Common Multiple
- Math formula
- Matrices
- Matrix Addition and Subtraction
- Matrix Determinant
- Matrix Multiplication
- Metric and Imperial Units
- Misleading Graphs
- Mixed Expressions
- Modulus Functions
- Modulus and Phase
- Multiples of Pi
- Multiplication and Division of Fractions
- Multiplicative Relationship
- Multiplying and Dividing Rational Expressions
- Natural Logarithm
- Natural Numbers
- Notation
- Number
- Number Line
- Number Systems
- Numerical Methods
- Odd functions
- Open Sentences and Identities
- Operation with Complex Numbers
- Operations with Decimals
- Operations with Matrices
- Operations with Polynomials
- Order of Operations
- Parabola
- Parallel Lines
- Parametric Differentiation
- Parametric Equations
- Parametric Integration
- Partial Fractions
- Pascal's Triangle
- Percentage
- Percentage Increase and Decrease
- Percentage as fraction or decimals
- Perimeter of a Triangle
- Permutations and Combinations
- Perpendicular Lines
- Points Lines and Planes
- Polynomial Graphs
- Polynomials
- Powers Roots And Radicals
- Powers and Exponents
- Powers and Roots
- Prime Factorization
- Prime Numbers
- Problem-solving Models and Strategies
- Product Rule
- Proof
- Proof and Mathematical Induction
- Proof by Contradiction
- Proof by Deduction
- Proof by Exhaustion
- Proof by Induction
- Properties of Exponents
- Proportion
- Proving an Identity
- Pythagorean Identities
- Quadratic Equations
- Quadratic Function Graphs
- Quadratic Graphs
- Quadratic functions
- Quadrilaterals
- Quotient Rule
- Radians
- Radical Functions
- Rates of Change
- Ratio
- Ratio Fractions
- Rational Exponents
- Rational Expressions
- Rational Functions
- Rational Numbers and Fractions
- Ratios as Fractions
- Real Numbers
- Reciprocal Graphs
- Recurrence Relation
- Recursion and Special Sequences
- Remainder and Factor Theorems
- Representation of Complex Numbers
- Rewriting Formulas and Equations
- Roots of Complex Numbers
- Roots of Polynomials
- Roots of Unity
- Rounding
- SAS Theorem
- SSS Theorem
- Scalar Triple Product
- Scale Drawings and Maps
- Scale Factors
- Scientific Notation
- Second Order Recurrence Relation
- Sector of a Circle
- Segment of a Circle
- Sequences
- Sequences and Series
- Series Maths
- Sets Math
- Similar Triangles
- Similar and Congruent Shapes
- Simple Interest
- Simplifying Fractions
- Simplifying Radicals
- Simultaneous Equations
- Sine and Cosine Rules
- Small Angle Approximation
- Solving Linear Equations
- Solving Linear Systems
- Solving Quadratic Equations
- Solving Radical Inequalities
- Solving Rational Equations
- Solving Simultaneous Equations Using Matrices
- Solving Systems of Inequalities
- Solving Trigonometric Equations
- Solving and Graphing Quadratic Equations
- Solving and Graphing Quadratic Inequalities
- Special Products
- Standard Form
- Standard Integrals
- Standard Unit
- Straight Line Graphs
- Substraction and addition of fractions
- Sum and Difference of Angles Formulas
- Sum of Natural Numbers
- Surds
- Surjective functions
- Tables and Graphs
- Tangent of a Circle
- The Quadratic Formula and the Discriminant
- Transformations
- Transformations of Graphs
- Translations of Trigonometric Functions
- Triangle Rules
- Triangle trigonometry
- Trigonometric Functions
- Trigonometric Functions of General Angles
- Trigonometric Identities
- Trigonometric Ratios
- Trigonometry
- Turning Points
- Types of Functions
- Types of Numbers
- Types of Triangles
- Unit Circle
- Units
- Variables in Algebra
- Vectors
- Verifying Trigonometric Identities
- Writing Equations
- Writing Linear Equations
- Statistics
- Bias in Experiments
- Binomial Distribution
- Binomial Hypothesis Test
- Bivariate Data
- Box Plots
- Categorical Data
- Categorical Variables
- Central Limit Theorem
- Chi Square Test for Goodness of Fit
- Chi Square Test for Homogeneity
- Chi Square Test for Independence
- Chi-Square Distribution
- Combining Random Variables
- Comparing Data
- Comparing Two Means Hypothesis Testing
- Conditional Probability
- Conducting a Study
- Conducting a Survey
- Conducting an Experiment
- Confidence Interval for Population Mean
- Confidence Interval for Population Proportion
- Confidence Interval for Slope of Regression Line
- Confidence Interval for the Difference of Two Means
- Confidence Intervals
- Correlation Math
- Cumulative Frequency
- Data Analysis
- Data Interpretation
- Discrete Random Variable
- Distributions
- Dot Plot
- Empirical Rule
- Errors in Hypothesis Testing
- Estimator Bias
- Events (Probability)
- Frequency Polygons
- Generalization and Conclusions
- Geometric Distribution
- Histograms
- Hypothesis Test for Correlation
- Hypothesis Test of Two Population Proportions
- Hypothesis Testing
- Inference for Distributions of Categorical Data
- Inferences in Statistics
- Large Data Set
- Least Squares Linear Regression
- Linear Interpolation
- Linear Regression
- Measures of Central Tendency
- Methods of Data Collection
- Normal Distribution
- Normal Distribution Hypothesis Test
- Normal Distribution Percentile
- Point Estimation
- Probability
- Probability Calculations
- Probability Distribution
- Probability Generating Function
- Quantitative Variables
- Quartiles
- Random Variables
- Randomized Block Design
- Residual Sum of Squares
- Residuals
- Sample Mean
- Sample Proportion
- Sampling
- Sampling Distribution
- Scatter Graphs
- Single Variable Data
- Skewness
- Standard Deviation
- Standard Normal Distribution
- Statistical Graphs
- Statistical Measures
- Stem and Leaf Graph
- Sum of Independent Random Variables
- Survey Bias
- Transforming Random Variables
- Tree Diagram
- Two Categorical Variables
- Two Quantitative Variables
- Type I Error
- Type II Error
- Types of Data in Statistics
- Venn Diagrams

What do the numbers 20 and 50 have in common? Well, both these numbers are divisible by 2, 5 and 10. We say this because no remainder will exist when we divide them by these three said numbers. This means that 20 and 50 are multiples of 2, 5 and 10 since

\[\mathbf{2}\times 10=\mathbf{5}\times 4=\mathbf{10}\times 2=20\]

\[\text{and}\]

\[\mathbf{2}\times 25=\mathbf{5}\times 10=\mathbf{10}\times 5=50\]

Looking at our derivation above, we can further infer that the numbers 2, 5 and 10 share two multiples, namely 20 and 50. These shared numbers are called **common multiples**. This article will demonstrate a method we can use to identify common multiples for a given set of numbers.

To ease ourselves into this topic, let us go through a quick overview of our previous topic on multiples.

A **multiple** of a non-zero integer \(A\) is a non-zero integer \(C\) that can be obtained by multiplying it with another number, say \(B\).

In other words, \(C\) is a multiple of \(A\) if \(C\) is in the multiplication table of \(A\).

The multiple of a number, say \(a\), is given by the general formula,

\[\text{Multiple of}\ a=a\times z\]

where \(z\in\mathbb{Z}\). In other words,

if \(A\times B=C\) then \(A\) and \(B\) are divisors (or factors) of \(C\),

**or **\(C\) is a multiple of \(A\) (and also \(B\)).

To find a particular set of multiples for a given number, we can use the multiplication table.

As with our example above, the numbers 20 and 50 are multiples of 2, 5 and 10. The following table shows other multiples of 2, 5 and 10.

First 6 non-zero multiples | |

2 | 2, 4, 6, 8, 10, 12 |

5 | 5, 10, 15, 20, 25, 30 |

20 | 20, 40, 60, 80, 100, 120 |

A more in-depth explanation of multiples can be found in the topic called Multiples.

Let us now define a common multiple.

A** common multiple** is a multiple that is shared between two (or more) numbers.

Identifying a common multiple(s) for a given set of numbers is fairly straightforward. Given a set of numbers, you would simply execute two steps:

** Step 1:** List the multiples of each number given in the set;

** Step 2**: Pick out any identical multiples shared from the lists written in Step 1.

Recall that there are an *infinite *number of multiples for any integer. With this property in mind, a restriction may be introduced in Step 1. In most cases, the question will define an interval for which the common multiples are satisfied for a given set of numbers.

For example, you may get questions that use the phrase "find the first two common multiples of 2 and 3" or "list the common multiples of 2 and 3 between 1 and 10". However, an interval restriction is not necessary here. But, it is safe to say that no individual can list all the common multiples for a given set of numbers by hand. That would take yards of ink and paper!

**Important note:** Although zero is indeed a common multiple for any set of numbers, you would typically list down the non-zero common multiples only (we shall look into this in the next section).

Here is an example for finding common multiples for a given set of numbers.

List all the (non-zero) common multiples of 9, 12 and 15 between 1 and 100.

**Solution**

The interval restriction here is that we need to list the multiples of 9, 12 and 15 between 1 and 100. We shall begin by listing these multiples using the table below.

Multiples between 1 and 100. | |

9 | 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99 |

12 | 12, 24, 36, 48, 60, 72, 84, 96 |

15 | 15, 30, 45, 60, 75, 90 |

Looking at the table above, there are no visible common multiples of 9, 12 and 15 for this interval. However, you can deduce the following ideas:

- The common multiples of 9 and 12 are 36 and 72 for this interval;
- The common multiples of 9 and 15 are 45 and 90 for this interval;
- The common multiple of 12 and 15 is 60 for this interval.

Here is another worked example.

List the first 2 non-zero common multiples of 5 and 17.

**Solution **

The interval restriction here is that we need to list the first 2 non-zero multiples of 5 and 17.

Sometimes, listing multiples can be rather cumbersome, especially when the numbers are very far apart from each other. As with our case here, the difference between 5 and 17 is quite large, so listing the multiples of 5 may take a while until we can find one that is also a multiple of 17.

For situations like this, it is advised to list down the multiples of the larger number and test whether these multiples are also multiples of the smaller number. We do this by verifying that it is divisible by each other (this will be further explained in the next section).

For this example, let us write down the first few non-zero multiples of 17.

**Multiples of 17:** 17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187,...

From this list, we can observe that 85 and 170 are indeed divisible by 5 since \(5\times 17=85\) and \(5\times 34=170\). Thus, the first 2 non-zero common multiples of 5 and 17 are 85 and 170.

Let us look at one more example before moving on to the next section.

List all the common multiples of 11 and 13 between 130 and 300.

**Solution **

The interval restriction here is that we need to list the multiples of 11 and 13 between 130 and 300. As before, we will start by listing these multiples using the table below.

Multiples between 130 and 300 | |

11 | 132, 143, 154, 165, 176, 187, 198, 209, 220, 231, 242, 253, 264, 275, 286, 297 |

13 | 130, 143, 156, 169, 182, 195, 208, 221, 234, 247, 260, 273, 286, 299 |

From the table above, observe that there are two common multiples of 11 and 13 between 130 and 300, namely 143 and 286.

The concept of a common multiple is primarily used to find the lowest common multiple (or LCM) between a given set of numbers. This is the smallest common multiple shared between two (or more) numbers. You can find a thorough discussion on this topic in the article: Lowest Common Multiple.

**Try it yourself: **Answer the following questions.

- What are the first two non-zero common multiples of 16 and 27?
- What are the common multiples of 9 and 12 between 22 and 140?

__Solutions__

Question 1: 432, 864

Question 2: 36, 72, 108

Before we move on to more examples involving common multiples, let us establish some important properties of common multiples.

Property | Example |

A set of numbers can have more than one common multiple. | 6 is a common multiple of 3 and 6. However, this is not the only common multiple of 3 and 6. The numbers 12, 18 and 24 are also some other common multiples of 3 and 6. |

A set of numbers can have an infinite number of common multiples. | Common multiples of 5 and 8 include 40, 80, 120, 160, ... The values will keep increasing and the list will go on forever. |

The common multiple of a set of numbers is always greater than or equal to each of the numbers themselves (excluding 0). | Common multiples of 2 and 4 between 1 and 10 are 4 and 8. Notice that the multiple 4 is greater than the given number 2 but equal to the given number 4. The multiple 8 however is greater than both the given numbers 2 and 4. |

The given set of numbers divides the common multiple without leaving a remainder. These numbers are called the factors. | A common multiple of 8 and 17 is 136. Dividing 136 by each of these given numbers will not produce a remainder since \(8\times 17=136\) and \(17\times 8=136\). |

Every non-zero integer is a multiple of 0 since any non-zero integer multiplied by 0 equal 0. In most cases, we will only consider non-zero common multiples. | Since \(7\times 0=0\) and \(9\times 0=0\) then 0 is a common multiple of 7 and 9. |

We shall end this topic by looking at a few more worked examples concerning common multiples.

List all the common multiples of 6, 8 and 10 between 1 and 100.

**Solution **

To begin, we shall list the multiples of each given number between 1 and 100. This is shown in the table below.

Multiples between 1 and 100 | |

6 | 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96 |

8 | 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96 |

10 | 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 |

From the table above, we find that there are no common multiples of 6, 8 and 10 between 1 and 100. However, we can indeed conclude the following relationships:

- The common multiples of 6 and 8 are 24, 48, 72 and 96 for this interval
- The common multiples of 6 and 10 are 30, 60 and 90 for this interval
- The common multiples of 8 and 10 are 4 and 80 for this interval

Let us now move on to another example.

List the first 4 non-zero common multiples of 2, 7 and 14.

**Solution**

First, note that 2 is relatively located far from 7 and 14 on the number line. Thus, it is sensible to list the multiples of 7 and 14 and compare their common multiples. From here, you will check whether these common multiples are also divisible by 2.

**Multiples of 7: **7, 14, 21, 28, 35, 42, 49, 56, 63, 70,...

**Multiples of 14: **14, 28, 42, 56, 70,...

Notice that the first 4 non-zero common multiples of 7 and 14 are 14, 28, 42 and 56. All four of these numbers are even which means that they are also divisible by 2. Thus, the first 4 non-zero common multiples of 2, 7 and 14 are 14, 28, 42 and 56.

We shall look at one last example involving common multiples.

List the first 3 non-zero common multiples of 3 and 19.

**Solution**

The difference between 3 and 19 is rather significant. So, as with our previous example, we will only list the multiples of 19 and verify whether they are also divisible by 3.

**Multiples of 19:** 19, 38, 57, 76, 95, 114, 133, 152, 171, 190,...

From this list, we find that the numbers 57, 114 and 171 are also divisible by 3 since \(3\times 19=57\), \(3\times 138=114\) and \(3\times 57=171\). Therefore, the first 3 non-zero common multiples of 3 and 19 are 57, 114 and 171.

Here is an interesting question: can we apply common multiples in real-life situations? As a matter of fact, we can! In this section, we shall demonstrate two examples of real-world scenarios that encapsulate all that we have learnt from this discussion.

Polly and Hannah decide to take turns visiting their friend, Ben, at the hospital. Polly suggests visiting Ben every 3 days while Hannah visits him every 5 days. If both of them visited Ben today, how long will it be until the next time they see him the same day again?

**Solution**

Here, we simply need to find the first* non-zero *common multiple of days 3 and 5. We shall take today as the first multiple of days 3 and 5 which is day 0. Remember, every non-zero integer is a multiple of 0 since any non-zero integer multiples by 0 equal 0 (property 5 of common multiples).

Let us now write down the common multiples of 3 and 5:

**Multiples of 3:** 2, 6, 9, 12, 15, 18,...

**Multiples of 5: **5, 10, 15, 20,...

From both these lists, we see that 15 is the first non-zero common multiple of 3 and 5. Hence, the next time both Polly and Hannah will visit Ben together is on day 15.

Here is the final real-world example to tie this article up.

Rory and Tana are jogging around a circular running track. Rory takes 12 minutes to complete a lap while Tana takes 16 minutes. If both of them leave the starting point at the same time, write down the next two times both of them will pass the starting point together again.

**Solution **

Using a similar approach as the previous example, we need to locate the first two* non-zero *common multiples of minutes 12 and 16. Taking the first time they leave the starting point says minute 0, we can now list the multiples of 12 and 16.

**Multiples of 12:** 12, 24, 36, 48, 60, 72, 84, 96, 108,...

**Multiples of 16:** 16, 32, 48, 64, 80, 96, 112,...

Looking at the lists above, notice that 48 and 96 are the first two non-zero common multiples of 12 and 16. Thus, Rory and Tana will pass the starting together again at minutes 48 and 96.

- A common multiple is a multiple that is shared between two numbers.
- To find the common multiples of a given set of numbers:
- List the multiples of each number given in the set;
- Pick out any identical multiples shared from the lists written in Step 1.

Important properties of common multiples:

A set of numbers can have more than one common multiple

A set of numbers can have an infinite number of common multiples

The common multiple of a set of numbers is always greater than or equal to each of the numbers themselves

The given set of numbers divides the common multiple without leaving a remainder.

Every non-zero integer is a multiple of 0 since any non-zero integer multiples by 0 equal 0.

A** **common multiple is a multiple that is shared between two (or more) numbers.

The characteristics/properties of common multiples are:

- a set of numbers can have more than one common multiple;
- a set of numbers can have an infinite number of common multiples;
- the common multiple of a set of numbers is always greater than or equal to each of the numbers themselves (excluding 0);
- the given set of numbers divides the common multiple without leaving a remainder;
- every non-zero integer is a multiple of 0 since any non-zero integer multiplied by 0 equal 0.

A common multiple is not to be solved but to be determined. And you can determine a common multiple of two (or more) numbers by executing two steps:

- Step 1: List the multiples of each number given in the set;
- Step 2: Pick out any identical multiples shared from the lists written in Step 1.

The rules of common multiples are the properties of common multiples.

Examples of common multiples of two (or more) number are:

- the first two non-zero common multiples of 16 and 27 are 432 and 864;
- the common multiples of 2, 7 and 14 between 1 and 60 are 14, 28, 42 and 56.

More about Common Multiples

Be perfectly prepared on time with an individual plan.

Test your knowledge with gamified quizzes.

Create and find flashcards in record time.

Create beautiful notes faster than ever before.

Have all your study materials in one place.

Upload unlimited documents and save them online.

Identify your study strength and weaknesses.

Set individual study goals and earn points reaching them.

Stop procrastinating with our study reminders.

Earn points, unlock badges and level up while studying.

Create flashcards in notes completely automatically.

Create the most beautiful study materials using our templates.

Sign up to highlight and take notes. It’s 100% free.