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LCM of 48, 56, and 72

LCM of 48, 56, and 72 is the smallest number among all common multiples of 48, 56, and 72. The first few multiples of 48, 56, and 72 are (48, 96, 144, 192, 240 . . .), (56, 112, 168, 224, 280 . . .), and (72, 144, 216, 288, 360 . . .) respectively. There are 3 commonly used methods to find LCM of 48, 56, 72 - by prime factorization, by listing multiples, and by division method.

1.	LCM of 48, 56, and 72
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 48, 56, and 72?

Answer: LCM of 48, 56, and 72 is 1008.

Explanation:

The LCM of three non-zero integers, a(48), b(56), and c(72), is the smallest positive integer m(1008) that is divisible by a(48), b(56), and c(72) without any remainder.

Methods to Find LCM of 48, 56, and 72

The methods to find the LCM of 48, 56, and 72 are explained below.

By Listing Multiples
By Prime Factorization Method
By Division Method

LCM of 48, 56, and 72 by Listing Multiples

To calculate the LCM of 48, 56, 72 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 48 (48, 96, 144, 192, 240 . . .), 56 (56, 112, 168, 224, 280 . . .), and 72 (72, 144, 216, 288, 360 . . .).
Step 2: The common multiples from the multiples of 48, 56, and 72 are 1008, 2016, . . .
Step 3: The smallest common multiple of 48, 56, and 72 is 1008.

∴ The least common multiple of 48, 56, and 72 = 1008.

LCM of 48, 56, and 72 by Prime Factorization

Prime factorization of 48, 56, and 72 is (2 × 2 × 2 × 2 × 3) = 2⁴ × 3¹, (2 × 2 × 2 × 7) = 2³ × 7¹, and (2 × 2 × 2 × 3 × 3) = 2³ × 3² respectively. LCM of 48, 56, and 72 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2⁴ × 3² × 7¹ = 1008.
Hence, the LCM of 48, 56, and 72 by prime factorization is 1008.

LCM of 48, 56, and 72 by Division Method

To calculate the LCM of 48, 56, and 72 by the division method, we will divide the numbers(48, 56, 72) by their prime factors (preferably common). The product of these divisors gives the LCM of 48, 56, and 72.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 48, 56, and 72. Write this prime number(2) on the left of the given numbers(48, 56, and 72), separated as per the ladder arrangement.
Step 2: If any of the given numbers (48, 56, 72) 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 48, 56, and 72 is the product of all prime numbers on the left, i.e. LCM(48, 56, 72) by division method = 2 × 2 × 2 × 2 × 3 × 3 × 7 = 1008.

☛ Also Check:

LCM of 48, 56, and 72 Examples

Example 1: Verify the relationship between the GCD and LCM of 48, 56, and 72.

Solution:

The relation between GCD and LCM of 48, 56, and 72 is given as,
LCM(48, 56, 72) = [(48 × 56 × 72) × GCD(48, 56, 72)]/[GCD(48, 56) × GCD(56, 72) × GCD(48, 72)]
⇒ Prime factorization of 48, 56 and 72:
- 48 = 2⁴ × 3¹
- 56 = 2³ × 7¹
- 72 = 2³ × 3²
∴ GCD of (48, 56), (56, 72), (48, 72) and (48, 56, 72) = 8, 8, 24 and 8 respectively.
Now, LHS = LCM(48, 56, 72) = 1008.
And, RHS = [(48 × 56 × 72) × GCD(48, 56, 72)]/[GCD(48, 56) × GCD(56, 72) × GCD(48, 72)] = [(193536) × 8]/[8 × 8 × 24] = 1008
LHS = RHS = 1008.
Hence verified.
Example 2: Find the smallest number that is divisible by 48, 56, 72 exactly.

Solution:

The value of LCM(48, 56, 72) will be the smallest number that is exactly divisible by 48, 56, and 72.
⇒ Multiples of 48, 56, and 72:
- Multiples of 48 = 48, 96, 144, 192, 240, 288, 336, 384, 432, 480, . . . ., 864, 912, 960, 1008, . . . .
- Multiples of 56 = 56, 112, 168, 224, 280, 336, 392, 448, 504, 560, . . . ., 784, 840, 896, 952, 1008, . . . .
- Multiples of 72 = 72, 144, 216, 288, 360, 432, 504, 576, 648, 720, . . . ., 864, 936, 1008, . . . .
Therefore, the LCM of 48, 56, and 72 is 1008.
Example 3: Calculate the LCM of 48, 56, and 72 using the GCD of the given numbers.

Solution:

Prime factorization of 48, 56, 72:
- 48 = 2⁴ × 3¹
- 56 = 2³ × 7¹
- 72 = 2³ × 3²
Therefore, GCD(48, 56) = 8, GCD(56, 72) = 8, GCD(48, 72) = 24, GCD(48, 56, 72) = 8
We know,
LCM(48, 56, 72) = [(48 × 56 × 72) × GCD(48, 56, 72)]/[GCD(48, 56) × GCD(56, 72) × GCD(48, 72)]
LCM(48, 56, 72) = (193536 × 8)/(8 × 8 × 24) = 1008
⇒LCM(48, 56, 72) = 1008

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FAQs on LCM of 48, 56, and 72

What is the LCM of 48, 56, and 72?

The LCM of 48, 56, and 72 is 1008. To find the LCM of 48, 56, and 72, we need to find the multiples of 48, 56, and 72 (multiples of 48 = 48, 96, 144, 192 . . . . 1008 . . . . ; multiples of 56 = 56, 112, 168, 224 . . . . 1008 . . . . ; multiples of 72 = 72, 144, 216, 288 . . . . 1008 . . . . ) and choose the smallest multiple that is exactly divisible by 48, 56, and 72, i.e., 1008.

What is the Least Perfect Square Divisible by 48, 56, and 72?

The least number divisible by 48, 56, and 72 = LCM(48, 56, 72)
LCM of 48, 56, and 72 = 2 × 2 × 2 × 2 × 3 × 3 × 7 [Incomplete pair(s): 7]
⇒ Least perfect square divisible by each 48, 56, and 72 = LCM(48, 56, 72) × 7 = 7056 [Square root of 7056 = √7056 = ±84]
Therefore, 7056 is the required number.

What are the Methods to Find LCM of 48, 56, 72?

The commonly used methods to find the LCM of 48, 56, 72 are:

Listing Multiples
Division Method
Prime Factorization Method

Which of the following is the LCM of 48, 56, and 72? 27, 25, 11, 1008

The value of LCM of 48, 56, 72 is the smallest common multiple of 48, 56, and 72. The number satisfying the given condition is 1008.

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