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LCM of 8, 15, and 21

LCM of 8, 15, and 21 is the smallest number among all common multiples of 8, 15, and 21. The first few multiples of 8, 15, and 21 are (8, 16, 24, 32, 40 . . .), (15, 30, 45, 60, 75 . . .), and (21, 42, 63, 84, 105 . . .) respectively. There are 3 commonly used methods to find LCM of 8, 15, 21 - by prime factorization, by division method, and by listing multiples.

1.	LCM of 8, 15, and 21
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 8, 15, and 21?

Answer: LCM of 8, 15, and 21 is 840.

Explanation:

The LCM of three non-zero integers, a(8), b(15), and c(21), is the smallest positive integer m(840) that is divisible by a(8), b(15), and c(21) without any remainder.

Methods to Find LCM of 8, 15, and 21

The methods to find the LCM of 8, 15, and 21 are explained below.

By Division Method
By Prime Factorization Method
By Listing Multiples

LCM of 8, 15, and 21 by Division Method

To calculate the LCM of 8, 15, and 21 by the division method, we will divide the numbers(8, 15, 21) by their prime factors (preferably common). The product of these divisors gives the LCM of 8, 15, and 21.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 8, 15, and 21. Write this prime number(2) on the left of the given numbers(8, 15, and 21), separated as per the ladder arrangement.
Step 2: If any of the given numbers (8, 15, 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, 15, and 21 is the product of all prime numbers on the left, i.e. LCM(8, 15, 21) by division method = 2 × 2 × 2 × 3 × 5 × 7 = 840.

LCM of 8, 15, and 21 by Prime Factorization

Prime factorization of 8, 15, and 21 is (2 × 2 × 2) = 2³, (3 × 5) = 3¹ × 5¹, and (3 × 7) = 3¹ × 7¹ respectively. LCM of 8, 15, and 21 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2³ × 3¹ × 5¹ × 7¹ = 840.
Hence, the LCM of 8, 15, and 21 by prime factorization is 840.

LCM of 8, 15, and 21 by Listing Multiples

To calculate the LCM of 8, 15, 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 . . .), 15 (15, 30, 45, 60, 75 . . .), and 21 (21, 42, 63, 84, 105 . . .).
Step 2: The common multiples from the multiples of 8, 15, and 21 are 840, 1680, . . .
Step 3: The smallest common multiple of 8, 15, and 21 is 840.

∴ The least common multiple of 8, 15, and 21 = 840.

☛ Also Check:

LCM of 8, 15, and 21 Examples

Example 1: Find the smallest number that is divisible by 8, 15, 21 exactly.

Solution:

The value of LCM(8, 15, 21) will be the smallest number that is exactly divisible by 8, 15, and 21.
⇒ Multiples of 8, 15, and 21:
- Multiples of 8 = 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, . . . ., 816, 824, 832, 840, . . . .
- Multiples of 15 = 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, . . . ., 810, 825, 840, . . . .
- Multiples of 21 = 21, 42, 63, 84, 105, 126, 147, 168, 189, 210, . . . ., 798, 819, 840, . . . .
Therefore, the LCM of 8, 15, and 21 is 840.
Example 2: Verify the relationship between the GCD and LCM of 8, 15, and 21.

Solution:

The relation between GCD and LCM of 8, 15, and 21 is given as,
LCM(8, 15, 21) = [(8 × 15 × 21) × GCD(8, 15, 21)]/[GCD(8, 15) × GCD(15, 21) × GCD(8, 21)]
⇒ Prime factorization of 8, 15 and 21:
- 8 = 2³
- 15 = 3¹ × 5¹
- 21 = 3¹ × 7¹
∴ GCD of (8, 15), (15, 21), (8, 21) and (8, 15, 21) = 1, 3, 1 and 1 respectively.
Now, LHS = LCM(8, 15, 21) = 840.
And, RHS = [(8 × 15 × 21) × GCD(8, 15, 21)]/[GCD(8, 15) × GCD(15, 21) × GCD(8, 21)] = [(2520) × 1]/[1 × 3 × 1] = 840
LHS = RHS = 840.
Hence verified.
Example 3: Calculate the LCM of 8, 15, and 21 using the GCD of the given numbers.

Solution:

Prime factorization of 8, 15, 21:
- 8 = 2³
- 15 = 3¹ × 5¹
- 21 = 3¹ × 7¹
Therefore, GCD(8, 15) = 1, GCD(15, 21) = 3, GCD(8, 21) = 1, GCD(8, 15, 21) = 1
We know,
LCM(8, 15, 21) = [(8 × 15 × 21) × GCD(8, 15, 21)]/[GCD(8, 15) × GCD(15, 21) × GCD(8, 21)]
LCM(8, 15, 21) = (2520 × 1)/(1 × 3 × 1) = 840
⇒LCM(8, 15, 21) = 840

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FAQs on LCM of 8, 15, and 21

What is the LCM of 8, 15, and 21?

The LCM of 8, 15, and 21 is 840. To find the least common multiple (LCM) of 8, 15, and 21, we need to find the multiples of 8, 15, and 21 (multiples of 8 = 8, 16, 24, 32 . . . . 840 . . . . ; multiples of 15 = 15, 30, 45, 60 . . . . 840 . . . . ; multiples of 21 = 21, 42, 63, 84 . . . . 840 . . . . ) and choose the smallest multiple that is exactly divisible by 8, 15, and 21, i.e., 840.

What is the Least Perfect Square Divisible by 8, 15, and 21?

The least number divisible by 8, 15, and 21 = LCM(8, 15, 21)
LCM of 8, 15, and 21 = 2 × 2 × 2 × 3 × 5 × 7 [Incomplete pair(s): 2, 3, 5, 7]
⇒ Least perfect square divisible by each 8, 15, and 21 = LCM(8, 15, 21) × 2 × 3 × 5 × 7 = 176400 [Square root of 176400 = √176400 = ±420]
Therefore, 176400 is the required number.

Which of the following is the LCM of 8, 15, and 21? 840, 15, 2, 10

The value of LCM of 8, 15, 21 is the smallest common multiple of 8, 15, and 21. The number satisfying the given condition is 840.

What is the Relation Between GCF and LCM of 8, 15, 21?

The following equation can be used to express the relation between GCF and LCM of 8, 15, 21, i.e. LCM(8, 15, 21) = [(8 × 15 × 21) × GCF(8, 15, 21)]/[GCF(8, 15) × GCF(15, 21) × GCF(8, 21)].

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