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101 LearnersLast updated on September 10, 2025
Properties of LCM
The least common multiple (LCM) is a mathematical concept that has several important properties. These properties help students simplify problems related to multiples and divisibility. The properties of LCM include its role in finding the smallest common multiple of given numbers and its relationship with the greatest common divisor (GCD). These properties help students analyze and solve problems related to fractions, ratios, and algebraic expressions. Now let us learn more about the properties of the LCM.
What are the Properties of LCM?
The properties of LCM are straightforward, and they help students to understand and work with multiples of numbers. These properties are derived from the principles of arithmetic. There are several properties of LCM, and some of them are mentioned below:
Property 1: Commutative Property
The LCM of two numbers is the same regardless of the order in which the numbers are considered.
Property 2: Associative Property
The LCM of three numbers can be found by first finding the LCM of any two numbers and then finding the LCM of the result with the third number.
Property 3: Relation with GCD
For any two numbers a and b, LCM(a, b) × GCD(a, b) = a × b.
Property 4: Multiplying by a Constant
If you multiply each of the numbers by a constant, the LCM of the resulting numbers is the original LCM multiplied by that constant.
Property 5: LCM of Prime Numbers
The LCM of two or more prime numbers is the product of the numbers themselves.
Tips and Tricks for Properties of LCM
Students tend to confuse and make mistakes while learning the properties of LCM. To avoid such confusion, we can follow the following tips and tricks:
Commutative and Associative Properties: Students should remember that the order in which numbers are considered does not affect the LCM. They should also practice grouping numbers to simplify calculations.
Relation with GCD: Students should remember that the LCM and GCD have a special relationship that can simplify calculations.
Multiplying by a Constant: Students should understand how multiplying numbers by a constant affects the LCM, as the result is proportionally scaled.
Confusing LCM with GCD
Students should remember that the LCM is the smallest common multiple while the GCD is the greatest common divisor. They should practice finding both to understand the differences.
Mistake 1
Misinterpreting the Commutative and Associative Properties
Students should know and remember that changing the order of numbers or grouping them differently does not affect the LCM.
Mistake 2
Incorrectly Applying the Relation with GCD
Students should practice using the relationship LCM(a, b) × GCD(a, b) = a × b to simplify calculations and understand the connection between the two concepts.
Mistake 3
Misunderstanding Multiplication by a Constant
Students should remember that multiplying numbers by a constant scales the LCM by that same constant.
Mistake 4
Forgetting the Prime Numbers Rule
Students must remember that the LCM of two or more prime numbers is simply their product, as they share no common factors.
Mistake 5
Solved Examples on the Properties of LCM
What is the LCM of 4 and 5?
LCM = 20
Problem 1
The LCM of two numbers is the smallest number that is a multiple of both. Here, 4 and 5 are prime to each other, so LCM = 4 × 5 = 20.
Find the LCM of 3, 6, and 9 using the associative property.
Explanation
LCM = 18
Problem 2
First, find the LCM of 3 and 6, which is 6. Then, find the LCM of 6 and 9, which is 18. Thus, the LCM of 3, 6, and 9 is 18.
If the LCM of 8 and 12 is 24, what is their GCD?
Explanation
GCD = 4
Problem 3
Using the relation LCM(a, b) × GCD(a, b) = a × b, we have 24 × GCD = 8 × 12. This simplifies to GCD = 96/24 = 4.
The LCM of 7 and 14 is 14. What can you conclude about 7 and 14?
Explanation
7 is a factor of 14.
Problem 4
Since 14 is the LCM, it indicates that one number is a multiple of the other. Here, 14 is a multiple of 7.
If you multiply 2 and 3 by 4, what is the LCM of the resulting numbers?
Explanation
LCM = 24
LCM is the least common multiple of two or more numbers, which is the smallest number that is a multiple of each of them.
1.How is LCM related to GCD?
2.Can the LCM of two numbers be smaller than both numbers?
3.Can LCM be used to simplify fractions?
4.How do you find the LCM of more than two numbers?
Common Mistakes and How to Avoid Them in Properties of LCM
Students tend to get confused when understanding the properties of LCM, and they tend to make mistakes while solving problems related to these properties. Here are some common mistakes students tend to make and the solutions to these common mistakes.
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Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
