LCM of 3,9 and 12
2026-02-28 09:47 Diff
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Last updated on August 5, 2025

The Least common multiple (LCM) is the smallest number that is divisible by the numbers 3,9 and 12. The LCM can be found using the listing multiples method, the prime factorization and/or division methods. LCM helps to solve problems with fractions and scenarios like scheduling or aligning repeating cycle of events.

What is the LCM of 3,9 and 12?

How to find the LCM of 3,9 and 12?

There are various methods to find the LCM, Listing method, prime factorization method and division method are explained below; 
 

LCM of 3,9 and 12 using the Listing Multiples Method

The LCM of 3,9 and 12 can be found using the following steps:


Step 1: Write down the multiples of each number


Multiples of 3  = 3,6,9,12,18,…36,…


Multiples of 9 = 9,18,27,36,…


Multiples of 12 = 12,24,36,…


Step 2:   Ascertain the smallest multiple from the listed multiples


The smallest common multiple is 36


Thus, LCM (3,9,12) = 36

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LCM of 3,9 and 12 using the Prime Factorization Method

The prime factors of each number are written, and then the highest power of the prime factors is multiplied to get the LCM.

Step 1: Find the prime factors of the numbers.

Prime factorization of 3 = 3

Prime factorization of 9 = 3×3

Prime factorization of 12 = 2×3×2

Step 2:Take the highest powers of each prime factor, and multiply the highest powers to get the LCM


3×3×2×2 =36


LCM(3,9, 12) = 36

LCM of 3,9 and 12 using the Division Method

This method involves dividing both numbers by their common prime factors until no further division is possible, then multiplying the divisors to find the LCM.

Steps:
Write the numbers, divide by common prime factors and multiply the divisors.

A prime integer that is evenly divisible into at least one of the provided numbers should be used to divide the row of numbers. 

Continue dividing the numbers until the last row of the results is ‘1’ and bring down the numbers not divisible by the previously chosen prime number. 

The LCM of the numbers is the product of the prime numbers in the first column, i.e., 


LCM (3,9,12) = 36
 

Common Mistakes and how to avoid them in LCM of 3,9 and 12

Listed below are a few commonly made mistakes while attempting to ascertain the LCM of 3,9 and 12, make a note while practicing.

Problem 1

LCM (3,9,x) = 36, find x.

Okay, lets begin

We know that; LCM (3,9,x) = 36 


The LCM of 3 and 9 = 9


Prime factorization of 36 = 32×42 


From the above prime factorization, we can assume that the missing factor must account for 22 or include 22 = 4

. 
The LCM includes the factor of 3 and 9 already, therefore x = 12. 
 

Explanation

 Making assumptions as above helps us to ascertain the missing number as explained above. 
  

Well explained 👍

Problem 2

Verify LCM(a,b,c)×HCF(a,b,c) = a×b×c , where a=3, b=9 and c=12.

Okay, lets begin

LCM of 3,9, 12;


Prime factorization of 3 = 3


Prime factorization of 9 = 3×3


Prime factorization of 12 = 2×3×2


LCM(3,9,12) = 36 


HCF of 3,9,12; 


Factors of 3 = 1,3


Factors of 9 = 1,3,9


Factors of 12 = 1,2,3,4,6,12 


HCF (3,9,12) = 3 


Verifying the above in the given formula; 


LCM(a,b,c)×HCF(a,b,c) = a×b×c


36×3 = 3×9×12


108 is not equal to 324

Explanation

The given formula doesn’t stand true when trying to verify for more than two given digits.
 

Well explained 👍

Problem 3

Find x, LCM(3,9,x) = 72

Okay, lets begin

We know that the LCM of 3,9 = 9 


The prime factorization of 72 = 23×32


The LCM of 3,9 already includes 32, the factor of x must include 23, which is 8.
 

Explanation

By following the above assumption we assume that the value of x is 8.

Well explained 👍

FAQs on LCM of 3,9 and 12

1.What is the LCM of 3,6,9,12?

Prime factorization of 3 = 3


Prime factorization of 6 = 2×3


Prime factorization of 9 = 3×3


Prime factorization of 12 = 2×3×2


LCM (3,6,9,12) = 36
 

2.What is the HCF of 3,9 and 12?

Factors of 3 = 1,3 


Factors of 9 = 1,3,9 


Factors of 12 = 1,2,3,4,6


HCF (3,9,12) = 3 
 

3.Find the LCM of 7,9 and 21.

Prime factorization of 7= 7


Prime factorization of 9 = 3×3


Prime factorization of 21 = 3×7


LCM (7,9,21) = 63
 

4.What is the LCM of 9 and 12?

Prime factorization of 9 = 3×3


Prime factorization of 12 = 2×3×2


LCM (9,12) = 36 
 

5.What is the LCM of 4,7 and 8?

Prime factorization of 4 = 2×2


Prime factorization of 7 = 7


Prime factorization of 8 =  2×2×2


LCM (4,7,8) = 56 
 

Important glossaries for the LCM of 3,9 and 12

  • Multiple: A number and any integer multiplied. 
  • Prime Factor: A natural number (other than 1) that has factors that are one and itself.
  • Prime Factorization: The process of breaking down a number into its prime factors is called Prime Factorization. 
  • Co-prime numbers: When the only positive integer that is a divisor of them both is 1, a number is co-prime. 
  • Relatively Prime Numbers: Numbers that have no common factors other than 1.
  • Fraction: A representation of a part of a whole.

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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.