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LCM of 28 and 32

LCM of 28 and 32 is the smallest number among all common multiples of 28 and 32. The first few multiples of 28 and 32 are (28, 56, 84, 112, 140, . . . ) and (32, 64, 96, 128, 160, 192, 224, . . . ) respectively. There are 3 commonly used methods to find LCM of 28 and 32 - by listing multiples, by division method, and by prime factorization.

1.	LCM of 28 and 32
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 28 and 32?

Answer: LCM of 28 and 32 is 224.

Explanation:

The LCM of two non-zero integers, x(28) and y(32), is the smallest positive integer m(224) that is divisible by both x(28) and y(32) without any remainder.

Methods to Find LCM of 28 and 32

Let's look at the different methods for finding the LCM of 28 and 32.

By Division Method
By Listing Multiples
By Prime Factorization Method

LCM of 28 and 32 by Division Method

To calculate the LCM of 28 and 32 by the division method, we will divide the numbers(28, 32) by their prime factors (preferably common). The product of these divisors gives the LCM of 28 and 32.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 28 and 32. Write this prime number(2) on the left of the given numbers(28 and 32), separated as per the ladder arrangement.
Step 2: If any of the given numbers (28, 32) 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 28 and 32 is the product of all prime numbers on the left, i.e. LCM(28, 32) by division method = 2 × 2 × 2 × 2 × 2 × 7 = 224.

LCM of 28 and 32 by Listing Multiples

To calculate the LCM of 28 and 32 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 28 (28, 56, 84, 112, 140, . . . ) and 32 (32, 64, 96, 128, 160, 192, 224, . . . . )
Step 2: The common multiples from the multiples of 28 and 32 are 224, 448, . . .
Step 3: The smallest common multiple of 28 and 32 is 224.

∴ The least common multiple of 28 and 32 = 224.

LCM of 28 and 32 by Prime Factorization

Prime factorization of 28 and 32 is (2 × 2 × 7) = 2² × 7¹ and (2 × 2 × 2 × 2 × 2) = 2⁵ respectively. LCM of 28 and 32 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2⁵ × 7¹ = 224.
Hence, the LCM of 28 and 32 by prime factorization is 224.

☛ Also Check:

LCM of 28 and 32 Examples

Example 1: Verify the relationship between GCF and LCM of 28 and 32.

Solution:

The relation between GCF and LCM of 28 and 32 is given as,
LCM(28, 32) × GCF(28, 32) = Product of 28, 32
Prime factorization of 28 and 32 is given as, 28 = (2 × 2 × 7) = 2² × 7¹ and 32 = (2 × 2 × 2 × 2 × 2) = 2⁵
LCM(28, 32) = 224
GCF(28, 32) = 4
LHS = LCM(28, 32) × GCF(28, 32) = 224 × 4 = 896
RHS = Product of 28, 32 = 28 × 32 = 896
⇒ LHS = RHS = 896
Hence, verified.
Example 2: The product of two numbers is 896. If their GCD is 4, what is their LCM?

Solution:

Given: GCD = 4
product of numbers = 896
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 896/4
Therefore, the LCM is 224.
The probable combination for the given case is LCM(28, 32) = 224.
Example 3: The GCD and LCM of two numbers are 4 and 224 respectively. If one number is 28, find the other number.

Solution:

Let the other number be z.
∵ GCD × LCM = 28 × z
⇒ z = (GCD × LCM)/28
⇒ z = (4 × 224)/28
⇒ z = 32
Therefore, the other number is 32.

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FAQs on LCM of 28 and 32

What is the LCM of 28 and 32?

The LCM of 28 and 32 is 224. To find the least common multiple (LCM) of 28 and 32, we need to find the multiples of 28 and 32 (multiples of 28 = 28, 56, 84, 112 . . . . 224; multiples of 32 = 32, 64, 96, 128 . . . . 224) and choose the smallest multiple that is exactly divisible by 28 and 32, i.e., 224.

If the LCM of 32 and 28 is 224, Find its GCF.

LCM(32, 28) × GCF(32, 28) = 32 × 28
Since the LCM of 32 and 28 = 224
⇒ 224 × GCF(32, 28) = 896
Therefore, the greatest common factor = 896/224 = 4.

Which of the following is the LCM of 28 and 32? 45, 224, 35, 5

The value of LCM of 28, 32 is the smallest common multiple of 28 and 32. The number satisfying the given condition is 224.

What is the Least Perfect Square Divisible by 28 and 32?

The least number divisible by 28 and 32 = LCM(28, 32)
LCM of 28 and 32 = 2 × 2 × 2 × 2 × 2 × 7 [Incomplete pair(s): 2, 7]
⇒ Least perfect square divisible by each 28 and 32 = LCM(28, 32) × 2 × 7 = 3136 [Square root of 3136 = √3136 = ±56]
Therefore, 3136 is the required number.

How to Find the LCM of 28 and 32 by Prime Factorization?

To find the LCM of 28 and 32 using prime factorization, we will find the prime factors, (28 = 2 × 2 × 7) and (32 = 2 × 2 × 2 × 2 × 2). LCM of 28 and 32 is the product of prime factors raised to their respective highest exponent among the numbers 28 and 32.
⇒ LCM of 28, 32 = 2⁵ × 7¹ = 224.

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