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LCM of 8 and 21

LCM of 8 and 21 is the smallest number among all common multiples of 8 and 21. The first few multiples of 8 and 21 are (8, 16, 24, 32, 40, 48, 56, . . . ) and (21, 42, 63, 84, 105, . . . ) respectively. There are 3 commonly used methods to find LCM of 8 and 21 - by division method, by prime factorization, and by listing multiples.

1. LCM of 8 and 21
2. List of Methods
3. Solved Examples
4. FAQs

What is the LCM of 8 and 21?

Answer: LCM of 8 and 21 is 168.

LCM of 8 and 21

Explanation:

The LCM of two non-zero integers, x(8) and y(21), is the smallest positive integer m(168) that is divisible by both x(8) and y(21) without any remainder.

Methods to Find LCM of 8 and 21

The methods to find the LCM of 8 and 21 are explained below.

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

LCM of 8 and 21 by Prime Factorization

Prime factorization of 8 and 21 is (2 × 2 × 2) = 23 and (3 × 7) = 31 × 71 respectively. LCM of 8 and 21 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 23 × 31 × 71 = 168.
Hence, the LCM of 8 and 21 by prime factorization is 168.

LCM of 8 and 21 by Division Method

To calculate the LCM of 8 and 21 by the division method, we will divide the numbers(8, 21) by their prime factors (preferably common). The product of these divisors gives the LCM of 8 and 21.

  • Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 8 and 21. Write this prime number(2) on the left of the given numbers(8 and 21), separated as per the ladder arrangement.
  • Step 2: If any of the given numbers (8, 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 8 and 21 is the product of all prime numbers on the left, i.e. LCM(8, 21) by division method = 2 × 2 × 2 × 3 × 7 = 168.

LCM of 8 and 21 by Listing Multiples

LCM of 8 and 21 by Listing Multiples Method

To calculate the LCM of 8 and 21 by listing out the common multiples, we can follow the given below steps:

  • Step 1: List a few multiples of 8 (8, 16, 24, 32, 40, 48, 56, . . . ) and 21 (21, 42, 63, 84, 105, . . . . )
  • Step 2: The common multiples from the multiples of 8 and 21 are 168, 336, . . .
  • Step 3: The smallest common multiple of 8 and 21 is 168.

∴ The least common multiple of 8 and 21 = 168.

☛ Also Check:

LCM of 8 and 21 Examples

  1. Example 1: Verify the relationship between GCF and LCM of 8 and 21.

    Solution:

    The relation between GCF and LCM of 8 and 21 is given as,
    LCM(8, 21) × GCF(8, 21) = Product of 8, 21
    Prime factorization of 8 and 21 is given as, 8 = (2 × 2 × 2) = 23 and 21 = (3 × 7) = 31 × 71
    LCM(8, 21) = 168
    GCF(8, 21) = 1
    LHS = LCM(8, 21) × GCF(8, 21) = 168 × 1 = 168
    RHS = Product of 8, 21 = 8 × 21 = 168
    ⇒ LHS = RHS = 168
    Hence, verified.

  2. Example 2: Find the smallest number that is divisible by 8 and 21 exactly.

    Solution:

    The value of LCM(8, 21) will be the smallest number that is exactly divisible by 8 and 21.
    ⇒ Multiples of 8 and 21:

    • Multiples of 8 = 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, . . . ., 136, 144, 152, 160, 168, . . . .
    • Multiples of 21 = 21, 42, 63, 84, 105, 126, 147, 168, 189, 210, . . . ., 84, 105, 126, 147, 168, . . . .

    Therefore, the LCM of 8 and 21 is 168.

  3. Example 3: The product of two numbers is 168. If their GCD is 1, what is their LCM?

    Solution:

    Given: GCD = 1
    product of numbers = 168
    ∵ LCM × GCD = product of numbers
    ⇒ LCM = Product/GCD = 168/1
    Therefore, the LCM is 168.
    The probable combination for the given case is LCM(8, 21) = 168.

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FAQs on LCM of 8 and 21

What is the LCM of 8 and 21?

The LCM of 8 and 21 is 168. To find the least common multiple (LCM) of 8 and 21, we need to find the multiples of 8 and 21 (multiples of 8 = 8, 16, 24, 32 . . . . 168; multiples of 21 = 21, 42, 63, 84 . . . . 168) and choose the smallest multiple that is exactly divisible by 8 and 21, i.e., 168.

Which of the following is the LCM of 8 and 21? 27, 24, 5, 168

The value of LCM of 8, 21 is the smallest common multiple of 8 and 21. The number satisfying the given condition is 168.

What are the Methods to Find LCM of 8 and 21?

The commonly used methods to find the LCM of 8 and 21 are:

  • Listing Multiples
  • Division Method
  • Prime Factorization Method

What is the Relation Between GCF and LCM of 8, 21?

The following equation can be used to express the relation between GCF and LCM of 8 and 21, i.e. GCF × LCM = 8 × 21.

If the LCM of 21 and 8 is 168, Find its GCF.

LCM(21, 8) × GCF(21, 8) = 21 × 8
Since the LCM of 21 and 8 = 168
⇒ 168 × GCF(21, 8) = 168
Therefore, the greatest common factor = 168/168 = 1.

 
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