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LCM of 378, 180, and 420

LCM of 378, 180, and 420 is the smallest number among all common multiples of 378, 180, and 420. The first few multiples of 378, 180, and 420 are (378, 756, 1134, 1512, 1890 . . .), (180, 360, 540, 720, 900 . . .), and (420, 840, 1260, 1680, 2100 . . .) respectively. There are 3 commonly used methods to find LCM of 378, 180, 420 - by prime factorization, by listing multiples, and by division method.

1. LCM of 378, 180, and 420
2. List of Methods
3. Solved Examples
4. FAQs

What is the LCM of 378, 180, and 420?

Answer: LCM of 378, 180, and 420 is 3780.

LCM of 378, 180, and 420

Explanation:

The LCM of three non-zero integers, a(378), b(180), and c(420), is the smallest positive integer m(3780) that is divisible by a(378), b(180), and c(420) without any remainder.

Methods to Find LCM of 378, 180, and 420

The methods to find the LCM of 378, 180, and 420 are explained below.

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

LCM of 378, 180, and 420 by Division Method

LCM of 378, 180, and 420 by Division Method

To calculate the LCM of 378, 180, and 420 by the division method, we will divide the numbers(378, 180, 420) by their prime factors (preferably common). The product of these divisors gives the LCM of 378, 180, and 420.

  • Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 378, 180, and 420. Write this prime number(2) on the left of the given numbers(378, 180, and 420), separated as per the ladder arrangement.
  • Step 2: If any of the given numbers (378, 180, 420) 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 378, 180, and 420 is the product of all prime numbers on the left, i.e. LCM(378, 180, 420) by division method = 2 × 2 × 3 × 3 × 3 × 5 × 7 = 3780.

LCM of 378, 180, and 420 by Listing Multiples

To calculate the LCM of 378, 180, 420 by listing out the common multiples, we can follow the given below steps:

  • Step 1: List a few multiples of 378 (378, 756, 1134, 1512, 1890 . . .), 180 (180, 360, 540, 720, 900 . . .), and 420 (420, 840, 1260, 1680, 2100 . . .).
  • Step 2: The common multiples from the multiples of 378, 180, and 420 are 3780, 7560, . . .
  • Step 3: The smallest common multiple of 378, 180, and 420 is 3780.

∴ The least common multiple of 378, 180, and 420 = 3780.

LCM of 378, 180, and 420 by Prime Factorization

Prime factorization of 378, 180, and 420 is (2 × 3 × 3 × 3 × 7) = 21 × 33 × 71, (2 × 2 × 3 × 3 × 5) = 22 × 32 × 51, and (2 × 2 × 3 × 5 × 7) = 22 × 31 × 51 × 71 respectively. LCM of 378, 180, and 420 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 22 × 33 × 51 × 71 = 3780.
Hence, the LCM of 378, 180, and 420 by prime factorization is 3780.

☛ Also Check:

LCM of 378, 180, and 420 Examples

  1. Example 1: Calculate the LCM of 378, 180, and 420 using the GCD of the given numbers.

    Solution:

    Prime factorization of 378, 180, 420:

    • 378 = 21 × 33 × 71
    • 180 = 22 × 32 × 51
    • 420 = 22 × 31 × 51 × 71

    Therefore, GCD(378, 180) = 18, GCD(180, 420) = 60, GCD(378, 420) = 42, GCD(378, 180, 420) = 6
    We know,
    LCM(378, 180, 420) = [(378 × 180 × 420) × GCD(378, 180, 420)]/[GCD(378, 180) × GCD(180, 420) × GCD(378, 420)]
    LCM(378, 180, 420) = (28576800 × 6)/(18 × 60 × 42) = 3780
    ⇒LCM(378, 180, 420) = 3780

  2. Example 2: Verify the relationship between the GCD and LCM of 378, 180, and 420.

    Solution:

    The relation between GCD and LCM of 378, 180, and 420 is given as,
    LCM(378, 180, 420) = [(378 × 180 × 420) × GCD(378, 180, 420)]/[GCD(378, 180) × GCD(180, 420) × GCD(378, 420)]
    ⇒ Prime factorization of 378, 180 and 420:

    • 378 = 21 × 33 × 71
    • 180 = 22 × 32 × 51
    • 420 = 22 × 31 × 51 × 71

    ∴ GCD of (378, 180), (180, 420), (378, 420) and (378, 180, 420) = 18, 60, 42 and 6 respectively.
    Now, LHS = LCM(378, 180, 420) = 3780.
    And, RHS = [(378 × 180 × 420) × GCD(378, 180, 420)]/[GCD(378, 180) × GCD(180, 420) × GCD(378, 420)] = [(28576800) × 6]/[18 × 60 × 42] = 3780
    LHS = RHS = 3780.
    Hence verified.

  3. Example 3: Find the smallest number that is divisible by 378, 180, 420 exactly.

    Solution:

    The value of LCM(378, 180, 420) will be the smallest number that is exactly divisible by 378, 180, and 420.
    ⇒ Multiples of 378, 180, and 420:

    • Multiples of 378 = 378, 756, 1134, 1512, 1890, 2268, 2646, 3024, 3402, 3780, . . . ., 3024, 3402, 3780, . . . .
    • Multiples of 180 = 180, 360, 540, 720, 900, 1080, 1260, 1440, 1620, 1800, . . . ., 3240, 3420, 3600, 3780, . . . .
    • Multiples of 420 = 420, 840, 1260, 1680, 2100, 2520, 2940, 3360, 3780, 4200, . . . ., 2520, 2940, 3360, 3780, . . . .

    Therefore, the LCM of 378, 180, and 420 is 3780.

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FAQs on LCM of 378, 180, and 420

What is the LCM of 378, 180, and 420?

The LCM of 378, 180, and 420 is 3780. To find the least common multiple of 378, 180, and 420, we need to find the multiples of 378, 180, and 420 (multiples of 378 = 378, 756, 1134, 1512 . . . . 3780 . . . . ; multiples of 180 = 180, 360, 540, 720 . . . . 3780 . . . . ; multiples of 420 = 420, 840, 1260, 1680 . . . . 3780 . . . . ) and choose the smallest multiple that is exactly divisible by 378, 180, and 420, i.e., 3780.

What is the Relation Between GCF and LCM of 378, 180, 420?

The following equation can be used to express the relation between GCF and LCM of 378, 180, 420, i.e. LCM(378, 180, 420) = [(378 × 180 × 420) × GCF(378, 180, 420)]/[GCF(378, 180) × GCF(180, 420) × GCF(378, 420)].

What are the Methods to Find LCM of 378, 180, 420?

The commonly used methods to find the LCM of 378, 180, 420 are:

  • Prime Factorization Method
  • Division Method
  • Listing Multiples

How to Find the LCM of 378, 180, and 420 by Prime Factorization?

To find the LCM of 378, 180, and 420 using prime factorization, we will find the prime factors, (378 = 21 × 33 × 71), (180 = 22 × 32 × 51), and (420 = 22 × 31 × 51 × 71). LCM of 378, 180, and 420 is the product of prime factors raised to their respective highest exponent among the numbers 378, 180, and 420.
⇒ LCM of 378, 180, 420 = 22 × 33 × 51 × 71 = 3780.

 
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