A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
A day full of math games & activities. Find one near you.
LearnPracticeDownload

LCM of 12, 24, and 36

LCM of 12, 24, and 36 is the smallest number among all common multiples of 12, 24, and 36. The first few multiples of 12, 24, and 36 are (12, 24, 36, 48, 60 . . .), (24, 48, 72, 96, 120 . . .), and (36, 72, 108, 144, 180 . . .) respectively. There are 3 commonly used methods to find LCM of 12, 24, 36 - by prime factorization, by division method, and by listing multiples.

1. LCM of 12, 24, and 36
2. List of Methods
3. Solved Examples
4. FAQs

What is the LCM of 12, 24, and 36?

Answer: LCM of 12, 24, and 36 is 72.

LCM of 12, 24, and 36

Explanation:

The LCM of three non-zero integers, a(12), b(24), and c(36), is the smallest positive integer m(72) that is divisible by a(12), b(24), and c(36) without any remainder.

Methods to Find LCM of 12, 24, and 36

The methods to find the LCM of 12, 24, and 36 are explained below.

  • By Division Method
  • By Listing Multiples
  • By Prime Factorization Method

LCM of 12, 24, and 36 by Division Method

LCM of 12, 24, and 36 by Division Method

To calculate the LCM of 12, 24, and 36 by the division method, we will divide the numbers(12, 24, 36) by their prime factors (preferably common). The product of these divisors gives the LCM of 12, 24, and 36.

  • Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 12, 24, and 36. Write this prime number(2) on the left of the given numbers(12, 24, and 36), separated as per the ladder arrangement.
  • Step 2: If any of the given numbers (12, 24, 36) is a multiple of 2, divide it by 2 and write the quotient below it. Bring down any number that is not divisible by the prime number.
  • Step 3: Continue the steps until only 1s are left in the last row.

The LCM of 12, 24, and 36 is the product of all prime numbers on the left, i.e. LCM(12, 24, 36) by division method = 2 × 2 × 2 × 3 × 3 = 72.

LCM of 12, 24, and 36 by Listing Multiples

LCM of 12, 24, and 36 by Listing Multiples Method

To calculate the LCM of 12, 24, 36 by listing out the common multiples, we can follow the given below steps:

  • Step 1: List a few multiples of 12 (12, 24, 36, 48, 60 . . .), 24 (24, 48, 72, 96, 120 . . .), and 36 (36, 72, 108, 144, 180 . . .).
  • Step 2: The common multiples from the multiples of 12, 24, and 36 are 72, 144, . . .
  • Step 3: The smallest common multiple of 12, 24, and 36 is 72.

∴ The least common multiple of 12, 24, and 36 = 72.

LCM of 12, 24, and 36 by Prime Factorization

Prime factorization of 12, 24, and 36 is (2 × 2 × 3) = 22 × 31, (2 × 2 × 2 × 3) = 23 × 31, and (2 × 2 × 3 × 3) = 22 × 32 respectively. LCM of 12, 24, and 36 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 23 × 32 = 72.
Hence, the LCM of 12, 24, and 36 by prime factorization is 72.

☛ Also Check:

LCM of 12, 24, and 36 Examples

  1. Example 1: Calculate the LCM of 12, 24, and 36 using the GCD of the given numbers.

    Solution:

    Prime factorization of 12, 24, 36:

    • 12 = 22 × 31
    • 24 = 23 × 31
    • 36 = 22 × 32

    Therefore, GCD(12, 24) = 12, GCD(24, 36) = 12, GCD(12, 36) = 12, GCD(12, 24, 36) = 12
    We know,
    LCM(12, 24, 36) = [(12 × 24 × 36) × GCD(12, 24, 36)]/[GCD(12, 24) × GCD(24, 36) × GCD(12, 36)]
    LCM(12, 24, 36) = (10368 × 12)/(12 × 12 × 12) = 72
    ⇒LCM(12, 24, 36) = 72

  2. Example 2: Find the smallest number that is divisible by 12, 24, 36 exactly.

    Solution:

    The smallest number that is divisible by 12, 24, and 36 exactly is their LCM.
    ⇒ Multiples of 12, 24, and 36:

    • Multiples of 12 = 12, 24, 36, 48, 60, 72, . . . .
    • Multiples of 24 = 24, 48, 72, 96, 120, 144, . . . .
    • Multiples of 36 = 36, 72, 108, 144, 180, 216, . . . .

    Therefore, the LCM of 12, 24, and 36 is 72.

  3. Example 3: Verify the relationship between the GCD and LCM of 12, 24, and 36.

    Solution:

    The relation between GCD and LCM of 12, 24, and 36 is given as,
    LCM(12, 24, 36) = [(12 × 24 × 36) × GCD(12, 24, 36)]/[GCD(12, 24) × GCD(24, 36) × GCD(12, 36)]
    ⇒ Prime factorization of 12, 24 and 36:

    • 12 = 22 × 31
    • 24 = 23 × 31
    • 36 = 22 × 32

    ∴ GCD of (12, 24), (24, 36), (12, 36) and (12, 24, 36) = 12, 12, 12 and 12 respectively.
    Now, LHS = LCM(12, 24, 36) = 72.
    And, RHS = [(12 × 24 × 36) × GCD(12, 24, 36)]/[GCD(12, 24) × GCD(24, 36) × GCD(12, 36)] = [(10368) × 12]/[12 × 12 × 12] = 72
    LHS = RHS = 72.
    Hence verified.

Show Solution >

go to slidego to slidego to slide

 

Ready to see the world through math’s eyes?
Math is at the core of everything we do. Enjoy solving real-world math problems in live classes and become an expert at everything. 

Book a Free Trial Class

FAQs on LCM of 12, 24, and 36

What is the LCM of 12, 24, and 36?

The LCM of 12, 24, and 36 is 72. To find the LCM (least common multiple) of 12, 24, and 36, we need to find the multiples of 12, 24, and 36 (multiples of 12 = 12, 24, 36, 48, 72 . . . .; multiples of 24 = 24, 48, 72, 96 . . . .; multiples of 36 = 36, 72, 108, 144 . . . .) and choose the smallest multiple that is exactly divisible by 12, 24, and 36, i.e., 72.

Which of the following is the LCM of 12, 24, and 36? 72, 3, 24, 10

The value of LCM of 12, 24, 36 is the smallest common multiple of 12, 24, and 36. The number satisfying the given condition is 72.

What is the Relation Between GCF and LCM of 12, 24, 36?

The following equation can be used to express the relation between GCF and LCM of 12, 24, 36, i.e. LCM(12, 24, 36) = [(12 × 24 × 36) × GCF(12, 24, 36)]/[GCF(12, 24) × GCF(24, 36) × GCF(12, 36)].

What are the Methods to Find LCM of 12, 24, 36?

The commonly used methods to find the LCM of 12, 24, 36 are:

  • Division Method
  • Prime Factorization Method
  • Listing Multiples
 
Download FREE Study Materials
LCM and GCF
Math worksheets and
visual curriculum