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LCM of 8, 9, and 10

LCM of 8, 9, and 10 is the smallest number among all common multiples of 8, 9, and 10. The first few multiples of 8, 9, and 10 are (8, 16, 24, 32, 40 . . .), (9, 18, 27, 36, 45 . . .), and (10, 20, 30, 40, 50 . . .) respectively. There are 3 commonly used methods to find LCM of 8, 9, 10 - by division method, by listing multiples, and by prime factorization.

1.	LCM of 8, 9, and 10
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 8, 9, and 10?

Answer: LCM of 8, 9, and 10 is 360.

Explanation:

The LCM of three non-zero integers, a(8), b(9), and c(10), is the smallest positive integer m(360) that is divisible by a(8), b(9), and c(10) without any remainder.

Methods to Find LCM of 8, 9, and 10

Let's look at the different methods for finding the LCM of 8, 9, and 10.

By Listing Multiples
By Prime Factorization Method
By Division Method

LCM of 8, 9, and 10 by Listing Multiples

To calculate the LCM of 8, 9, 10 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 . . .), 9 (9, 18, 27, 36, 45 . . .), and 10 (10, 20, 30, 40, 50 . . .).
Step 2: The common multiples from the multiples of 8, 9, and 10 are 360, 720, . . .
Step 3: The smallest common multiple of 8, 9, and 10 is 360.

∴ The least common multiple of 8, 9, and 10 = 360.

LCM of 8, 9, and 10 by Prime Factorization

Prime factorization of 8, 9, and 10 is (2 × 2 × 2) = 2³, (3 × 3) = 3², and (2 × 5) = 2¹ × 5¹ respectively. LCM of 8, 9, and 10 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2³ × 3² × 5¹ = 360.
Hence, the LCM of 8, 9, and 10 by prime factorization is 360.

LCM of 8, 9, and 10 by Division Method

To calculate the LCM of 8, 9, and 10 by the division method, we will divide the numbers(8, 9, 10) by their prime factors (preferably common). The product of these divisors gives the LCM of 8, 9, and 10.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 8, 9, and 10. Write this prime number(2) on the left of the given numbers(8, 9, and 10), separated as per the ladder arrangement.
Step 2: If any of the given numbers (8, 9, 10) 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, 9, and 10 is the product of all prime numbers on the left, i.e. LCM(8, 9, 10) by division method = 2 × 2 × 2 × 3 × 3 × 5 = 360.

☛ Also Check:

LCM of 8, 9, and 10 Examples

Example 1: Calculate the LCM of 8, 9, and 10 using the GCD of the given numbers.

Solution:

Prime factorization of 8, 9, 10:
- 8 = 2³
- 9 = 3²
- 10 = 2¹ × 5¹
Therefore, GCD(8, 9) = 1, GCD(9, 10) = 1, GCD(8, 10) = 2, GCD(8, 9, 10) = 1
We know,
LCM(8, 9, 10) = [(8 × 9 × 10) × GCD(8, 9, 10)]/[GCD(8, 9) × GCD(9, 10) × GCD(8, 10)]
LCM(8, 9, 10) = (720 × 1)/(1 × 1 × 2) = 360
⇒LCM(8, 9, 10) = 360
Example 2: Find the smallest number that is divisible by 8, 9, 10 exactly.

Solution:

The value of LCM(8, 9, 10) will be the smallest number that is exactly divisible by 8, 9, and 10.
⇒ Multiples of 8, 9, and 10:
- Multiples of 8 = 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, . . . ., 328, 336, 344, 352, 360, . . . .
- Multiples of 9 = 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, . . . ., 324, 333, 342, 351, 360, . . . .
- Multiples of 10 = 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, . . . ., 320, 330, 340, 350, 360, . . . .
Therefore, the LCM of 8, 9, and 10 is 360.
Example 3: Verify the relationship between the GCD and LCM of 8, 9, and 10.

Solution:

The relation between GCD and LCM of 8, 9, and 10 is given as,
LCM(8, 9, 10) = [(8 × 9 × 10) × GCD(8, 9, 10)]/[GCD(8, 9) × GCD(9, 10) × GCD(8, 10)]
⇒ Prime factorization of 8, 9 and 10:
- 8 = 2³
- 9 = 3²
- 10 = 2¹ × 5¹
∴ GCD of (8, 9), (9, 10), (8, 10) and (8, 9, 10) = 1, 1, 2 and 1 respectively.
Now, LHS = LCM(8, 9, 10) = 360.
And, RHS = [(8 × 9 × 10) × GCD(8, 9, 10)]/[GCD(8, 9) × GCD(9, 10) × GCD(8, 10)] = [(720) × 1]/[1 × 1 × 2] = 360
LHS = RHS = 360.
Hence verified.

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FAQs on LCM of 8, 9, and 10

What is the LCM of 8, 9, and 10?

The LCM of 8, 9, and 10 is 360. To find the least common multiple (LCM) of 8, 9, and 10, we need to find the multiples of 8, 9, and 10 (multiples of 8 = 8, 16, 24, 32 . . . . 360 . . . . ; multiples of 9 = 9, 18, 27, 36 . . . . 360 . . . . ; multiples of 10 = 10, 20, 30, 40 . . . . 360 . . . . ) and choose the smallest multiple that is exactly divisible by 8, 9, and 10, i.e., 360.

Which of the following is the LCM of 8, 9, and 10? 52, 10, 42, 360

The value of LCM of 8, 9, 10 is the smallest common multiple of 8, 9, and 10. The number satisfying the given condition is 360.

What is the Least Perfect Square Divisible by 8, 9, and 10?

The least number divisible by 8, 9, and 10 = LCM(8, 9, 10)
LCM of 8, 9, and 10 = 2 × 2 × 2 × 3 × 3 × 5 [Incomplete pair(s): 2, 5]
⇒ Least perfect square divisible by each 8, 9, and 10 = LCM(8, 9, 10) × 2 × 5 = 3600 [Square root of 3600 = √3600 = ±60]
Therefore, 3600 is the required number.

What is the Relation Between GCF and LCM of 8, 9, 10?

The following equation can be used to express the relation between GCF and LCM of 8, 9, 10, i.e. LCM(8, 9, 10) = [(8 × 9 × 10) × GCF(8, 9, 10)]/[GCF(8, 9) × GCF(9, 10) × GCF(8, 10)].

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