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LCM of 6 and 10

LCM of 6 and 10 is the smallest number among all common multiples of 6 and 10. The first few multiples of 6 and 10 are (6, 12, 18, 24, 30, 36, 42, . . . ) and (10, 20, 30, 40, . . . ) respectively. There are 3 commonly used methods to find LCM of 6 and 10 - by listing multiples, by prime factorization, and by division method.

1.	LCM of 6 and 10
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 6 and 10?

Answer: LCM of 6 and 10 is 30.

Explanation:

The LCM of two non-zero integers, x(6) and y(10), is the smallest positive integer m(30) that is divisible by both x(6) and y(10) without any remainder.

Methods to Find LCM of 6 and 10

The methods to find the LCM of 6 and 10 are explained below.

By Division Method
By Listing Multiples
By Prime Factorization Method

LCM of 6 and 10 by Division Method

To calculate the LCM of 6 and 10 by the division method, we will divide the numbers(6, 10) by their prime factors (preferably common). The product of these divisors gives the LCM of 6 and 10.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 6 and 10. Write this prime number(2) on the left of the given numbers(6 and 10), separated as per the ladder arrangement.
Step 2: If any of the given numbers (6, 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 6 and 10 is the product of all prime numbers on the left, i.e. LCM(6, 10) by division method = 2 × 3 × 5 = 30.

LCM of 6 and 10 by Listing Multiples

To calculate the LCM of 6 and 10 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 6 (6, 12, 18, 24, 30, 36, 42, . . . ) and 10 (10, 20, 30, 40, . . . . )
Step 2: The common multiples from the multiples of 6 and 10 are 30, 60, . . .
Step 3: The smallest common multiple of 6 and 10 is 30.

∴ The least common multiple of 6 and 10 = 30.

LCM of 6 and 10 by Prime Factorization

Prime factorization of 6 and 10 is (2 × 3) = 2¹ × 3¹ and (2 × 5) = 2¹ × 5¹ respectively. LCM of 6 and 10 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2¹ × 3¹ × 5¹ = 30.
Hence, the LCM of 6 and 10 by prime factorization is 30.

☛ Also Check:

LCM of 6 and 10 Examples

Example 1: Find the smallest number that is divisible by 6 and 10 exactly.

Solution:

The smallest number that is divisible by 6 and 10 exactly is their LCM.
⇒ Multiples of 6 and 10:
- Multiples of 6 = 6, 12, 18, 24, 30, 36, . . . .
- Multiples of 10 = 10, 20, 30, 40, 50, 60, . . . .
Therefore, the LCM of 6 and 10 is 30.
Example 2: Verify the relationship between GCF and LCM of 6 and 10.

Solution:

The relation between GCF and LCM of 6 and 10 is given as,
LCM(6, 10) × GCF(6, 10) = Product of 6, 10
Prime factorization of 6 and 10 is given as, 6 = (2 × 3) = 2¹ × 3¹ and 10 = (2 × 5) = 2¹ × 5¹
LCM(6, 10) = 30
GCF(6, 10) = 2
LHS = LCM(6, 10) × GCF(6, 10) = 30 × 2 = 60
RHS = Product of 6, 10 = 6 × 10 = 60
⇒ LHS = RHS = 60
Hence, verified.
Example 3: The product of two numbers is 60. If their GCD is 2, what is their LCM?

Solution:

Given: GCD = 2
product of numbers = 60
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 60/2
Therefore, the LCM is 30.
The probable combination for the given case is LCM(6, 10) = 30.

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FAQs on LCM of 6 and 10

What is the LCM of 6 and 10?

The LCM of 6 and 10 is 30. To find the LCM (least common multiple) of 6 and 10, we need to find the multiples of 6 and 10 (multiples of 6 = 6, 12, 18, 24 . . . . 30; multiples of 10 = 10, 20, 30, 40) and choose the smallest multiple that is exactly divisible by 6 and 10, i.e., 30.

What is the Least Perfect Square Divisible by 6 and 10?

The least number divisible by 6 and 10 = LCM(6, 10)
LCM of 6 and 10 = 2 × 3 × 5 [Incomplete pair(s): 2, 3, 5]
⇒ Least perfect square divisible by each 6 and 10 = LCM(6, 10) × 2 × 3 × 5 = 900 [Square root of 900 = √900 = ±30]
Therefore, 900 is the required number.

If the LCM of 10 and 6 is 30, Find its GCF.

LCM(10, 6) × GCF(10, 6) = 10 × 6
Since the LCM of 10 and 6 = 30
⇒ 30 × GCF(10, 6) = 60
Therefore, the greatest common factor (GCF) = 60/30 = 2.

What is the Relation Between GCF and LCM of 6, 10?

The following equation can be used to express the relation between GCF and LCM of 6 and 10, i.e. GCF × LCM = 6 × 10.

Which of the following is the LCM of 6 and 10? 30, 42, 11, 5

The value of LCM of 6, 10 is the smallest common multiple of 6 and 10. The number satisfying the given condition is 30.

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