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LCM of 12, 30, and 42

LCM of 12, 30, and 42 is the smallest number among all common multiples of 12, 30, and 42. The first few multiples of 12, 30, and 42 are (12, 24, 36, 48, 60 . . .), (30, 60, 90, 120, 150 . . .), and (42, 84, 126, 168, 210 . . .) respectively. There are 3 commonly used methods to find LCM of 12, 30, 42 - by division method, by listing multiples, and by prime factorization.

1.	LCM of 12, 30, and 42
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 12, 30, and 42?

Answer: LCM of 12, 30, and 42 is 420.

Explanation:

The LCM of three non-zero integers, a(12), b(30), and c(42), is the smallest positive integer m(420) that is divisible by a(12), b(30), and c(42) without any remainder.

Methods to Find LCM of 12, 30, and 42

The methods to find the LCM of 12, 30, and 42 are explained below.

By Listing Multiples
By Prime Factorization Method
By Division Method

LCM of 12, 30, and 42 by Listing Multiples

To calculate the LCM of 12, 30, 42 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 12 (12, 24, 36, 48, 60 . . .), 30 (30, 60, 90, 120, 150 . . .), and 42 (42, 84, 126, 168, 210 . . .).
Step 2: The common multiples from the multiples of 12, 30, and 42 are 420, 840, . . .
Step 3: The smallest common multiple of 12, 30, and 42 is 420.

∴ The least common multiple of 12, 30, and 42 = 420.

LCM of 12, 30, and 42 by Prime Factorization

Prime factorization of 12, 30, and 42 is (2 × 2 × 3) = 2² × 3¹, (2 × 3 × 5) = 2¹ × 3¹ × 5¹, and (2 × 3 × 7) = 2¹ × 3¹ × 7¹ respectively. LCM of 12, 30, and 42 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2² × 3¹ × 5¹ × 7¹ = 420.
Hence, the LCM of 12, 30, and 42 by prime factorization is 420.

LCM of 12, 30, and 42 by Division Method

To calculate the LCM of 12, 30, and 42 by the division method, we will divide the numbers(12, 30, 42) by their prime factors (preferably common). The product of these divisors gives the LCM of 12, 30, and 42.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 12, 30, and 42. Write this prime number(2) on the left of the given numbers(12, 30, and 42), separated as per the ladder arrangement.
Step 2: If any of the given numbers (12, 30, 42) 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 12, 30, and 42 is the product of all prime numbers on the left, i.e. LCM(12, 30, 42) by division method = 2 × 2 × 3 × 5 × 7 = 420.

☛ Also Check:

LCM of 12, 30, and 42 Examples

Example 1: Calculate the LCM of 12, 30, and 42 using the GCD of the given numbers.

Solution:

Prime factorization of 12, 30, 42:
- 12 = 2² × 3¹
- 30 = 2¹ × 3¹ × 5¹
- 42 = 2¹ × 3¹ × 7¹
Therefore, GCD(12, 30) = 6, GCD(30, 42) = 6, GCD(12, 42) = 6, GCD(12, 30, 42) = 6
We know,
LCM(12, 30, 42) = [(12 × 30 × 42) × GCD(12, 30, 42)]/[GCD(12, 30) × GCD(30, 42) × GCD(12, 42)]
LCM(12, 30, 42) = (15120 × 6)/(6 × 6 × 6) = 420
⇒LCM(12, 30, 42) = 420
Example 2: Find the smallest number that is divisible by 12, 30, 42 exactly.

Solution:

The value of LCM(12, 30, 42) will be the smallest number that is exactly divisible by 12, 30, and 42.
⇒ Multiples of 12, 30, and 42:
- Multiples of 12 = 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, . . . ., 396, 408, 420, . . . .
- Multiples of 30 = 30, 60, 90, 120, 150, 180, 210, 240, 270, 300, . . . ., 330, 360, 390, 420, . . . .
- Multiples of 42 = 42, 84, 126, 168, 210, 252, 294, 336, 378, 420, . . . ., 336, 378, 420, . . . .
Therefore, the LCM of 12, 30, and 42 is 420.
Example 3: Verify the relationship between the GCD and LCM of 12, 30, and 42.

Solution:

The relation between GCD and LCM of 12, 30, and 42 is given as,
LCM(12, 30, 42) = [(12 × 30 × 42) × GCD(12, 30, 42)]/[GCD(12, 30) × GCD(30, 42) × GCD(12, 42)]
⇒ Prime factorization of 12, 30 and 42:
- 12 = 2² × 3¹
- 30 = 2¹ × 3¹ × 5¹
- 42 = 2¹ × 3¹ × 7¹
∴ GCD of (12, 30), (30, 42), (12, 42) and (12, 30, 42) = 6, 6, 6 and 6 respectively.
Now, LHS = LCM(12, 30, 42) = 420.
And, RHS = [(12 × 30 × 42) × GCD(12, 30, 42)]/[GCD(12, 30) × GCD(30, 42) × GCD(12, 42)] = [(15120) × 6]/[6 × 6 × 6] = 420
LHS = RHS = 420.
Hence verified.

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FAQs on LCM of 12, 30, and 42

What is the LCM of 12, 30, and 42?

The LCM of 12, 30, and 42 is 420. To find the least common multiple (LCM) of 12, 30, and 42, we need to find the multiples of 12, 30, and 42 (multiples of 12 = 12, 24, 36, 48 . . . . 420 . . . . ; multiples of 30 = 30, 60, 90, 120 . . . . 420 . . . . ; multiples of 42 = 42, 84, 126, 168 . . . . 420 . . . . ) and choose the smallest multiple that is exactly divisible by 12, 30, and 42, i.e., 420.

Which of the following is the LCM of 12, 30, and 42? 420, 18, 110, 28

The value of LCM of 12, 30, 42 is the smallest common multiple of 12, 30, and 42. The number satisfying the given condition is 420.

What is the Relation Between GCF and LCM of 12, 30, 42?

The following equation can be used to express the relation between GCF and LCM of 12, 30, 42, i.e. LCM(12, 30, 42) = [(12 × 30 × 42) × GCF(12, 30, 42)]/[GCF(12, 30) × GCF(30, 42) × GCF(12, 42)].

What are the Methods to Find LCM of 12, 30, 42?

The commonly used methods to find the LCM of 12, 30, 42 are:

Prime Factorization Method
Listing Multiples
Division Method

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