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 9, 14, and 21

LCM of 9, 14, and 21 is the smallest number among all common multiples of 9, 14, and 21. The first few multiples of 9, 14, and 21 are (9, 18, 27, 36, 45 . . .), (14, 28, 42, 56, 70 . . .), and (21, 42, 63, 84, 105 . . .) respectively. There are 3 commonly used methods to find LCM of 9, 14, 21 - by prime factorization, by division method, and by listing multiples.

1. LCM of 9, 14, and 21
2. List of Methods
3. Solved Examples
4. FAQs

What is the LCM of 9, 14, and 21?

Answer: LCM of 9, 14, and 21 is 126.

LCM of 9, 14, and 21

Explanation:

The LCM of three non-zero integers, a(9), b(14), and c(21), is the smallest positive integer m(126) that is divisible by a(9), b(14), and c(21) without any remainder.

Methods to Find LCM of 9, 14, and 21

Let's look at the different methods for finding the LCM of 9, 14, and 21.

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

LCM of 9, 14, and 21 by Division Method

LCM of 9, 14, and 21 by Division Method

To calculate the LCM of 9, 14, and 21 by the division method, we will divide the numbers(9, 14, 21) by their prime factors (preferably common). The product of these divisors gives the LCM of 9, 14, and 21.

  • Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 9, 14, and 21. Write this prime number(2) on the left of the given numbers(9, 14, and 21), separated as per the ladder arrangement.
  • Step 2: If any of the given numbers (9, 14, 21) 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 9, 14, and 21 is the product of all prime numbers on the left, i.e. LCM(9, 14, 21) by division method = 2 × 3 × 3 × 7 = 126.

LCM of 9, 14, and 21 by Prime Factorization

Prime factorization of 9, 14, and 21 is (3 × 3) = 32, (2 × 7) = 21 × 71, and (3 × 7) = 31 × 71 respectively. LCM of 9, 14, and 21 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 21 × 32 × 71 = 126.
Hence, the LCM of 9, 14, and 21 by prime factorization is 126.

LCM of 9, 14, and 21 by Listing Multiples

To calculate the LCM of 9, 14, 21 by listing out the common multiples, we can follow the given below steps:

  • Step 1: List a few multiples of 9 (9, 18, 27, 36, 45 . . .), 14 (14, 28, 42, 56, 70 . . .), and 21 (21, 42, 63, 84, 105 . . .).
  • Step 2: The common multiples from the multiples of 9, 14, and 21 are 126, 252, . . .
  • Step 3: The smallest common multiple of 9, 14, and 21 is 126.

∴ The least common multiple of 9, 14, and 21 = 126.

☛ Also Check:

LCM of 9, 14, and 21 Examples

  1. Example 1: Calculate the LCM of 9, 14, and 21 using the GCD of the given numbers.

    Solution:

    Prime factorization of 9, 14, 21:

    • 9 = 32
    • 14 = 21 × 71
    • 21 = 31 × 71

    Therefore, GCD(9, 14) = 1, GCD(14, 21) = 7, GCD(9, 21) = 3, GCD(9, 14, 21) = 1
    We know,
    LCM(9, 14, 21) = [(9 × 14 × 21) × GCD(9, 14, 21)]/[GCD(9, 14) × GCD(14, 21) × GCD(9, 21)]
    LCM(9, 14, 21) = (2646 × 1)/(1 × 7 × 3) = 126
    ⇒LCM(9, 14, 21) = 126

  2. Example 2: Verify the relationship between the GCD and LCM of 9, 14, and 21.

    Solution:

    The relation between GCD and LCM of 9, 14, and 21 is given as,
    LCM(9, 14, 21) = [(9 × 14 × 21) × GCD(9, 14, 21)]/[GCD(9, 14) × GCD(14, 21) × GCD(9, 21)]
    ⇒ Prime factorization of 9, 14 and 21:

    • 9 = 32
    • 14 = 21 × 71
    • 21 = 31 × 71

    ∴ GCD of (9, 14), (14, 21), (9, 21) and (9, 14, 21) = 1, 7, 3 and 1 respectively.
    Now, LHS = LCM(9, 14, 21) = 126.
    And, RHS = [(9 × 14 × 21) × GCD(9, 14, 21)]/[GCD(9, 14) × GCD(14, 21) × GCD(9, 21)] = [(2646) × 1]/[1 × 7 × 3] = 126
    LHS = RHS = 126.
    Hence verified.

  3. Example 3: Find the smallest number that is divisible by 9, 14, 21 exactly.

    Solution:

    The smallest number that is divisible by 9, 14, and 21 exactly is their LCM.
    ⇒ Multiples of 9, 14, and 21:

    • Multiples of 9 = 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, . . . .
    • Multiples of 14 = 14, 28, 42, 56, 70, 84, 98, 112, 126, . . . .
    • Multiples of 21 = 21, 42, 63, 84, 105, 126, 147, . . . .

    Therefore, the LCM of 9, 14, and 21 is 126.

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 9, 14, and 21

What is the LCM of 9, 14, and 21?

The LCM of 9, 14, and 21 is 126. To find the LCM of 9, 14, and 21, we need to find the multiples of 9, 14, and 21 (multiples of 9 = 9, 18, 27, 36 . . . . 126 . . . . ; multiples of 14 = 14, 28, 42, 56 . . . . 126 . . . . ; multiples of 21 = 21, 42, 63, 84, 126 . . . .) and choose the smallest multiple that is exactly divisible by 9, 14, and 21, i.e., 126.

Which of the following is the LCM of 9, 14, and 21? 126, 21, 11, 110

The value of LCM of 9, 14, 21 is the smallest common multiple of 9, 14, and 21. The number satisfying the given condition is 126.

How to Find the LCM of 9, 14, and 21 by Prime Factorization?

To find the LCM of 9, 14, and 21 using prime factorization, we will find the prime factors, (9 = 32), (14 = 21 × 71), and (21 = 31 × 71). LCM of 9, 14, and 21 is the product of prime factors raised to their respective highest exponent among the numbers 9, 14, and 21.
⇒ LCM of 9, 14, 21 = 21 × 32 × 71 = 126.

What are the Methods to Find LCM of 9, 14, 21?

The commonly used methods to find the LCM of 9, 14, 21 are:

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