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LCM of 28, 36, and 45

LCM of 28, 36, and 45 is the smallest number among all common multiples of 28, 36, and 45. The first few multiples of 28, 36, and 45 are (28, 56, 84, 112, 140 . . .), (36, 72, 108, 144, 180 . . .), and (45, 90, 135, 180, 225 . . .) respectively. There are 3 commonly used methods to find LCM of 28, 36, 45 - by division method, by prime factorization, and by listing multiples.

1.	LCM of 28, 36, and 45
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 28, 36, and 45?

Answer: LCM of 28, 36, and 45 is 1260.

Explanation:

The LCM of three non-zero integers, a(28), b(36), and c(45), is the smallest positive integer m(1260) that is divisible by a(28), b(36), and c(45) without any remainder.

Methods to Find LCM of 28, 36, and 45

The methods to find the LCM of 28, 36, and 45 are explained below.

By Listing Multiples
By Division Method
By Prime Factorization Method

LCM of 28, 36, and 45 by Listing Multiples

To calculate the LCM of 28, 36, 45 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 . . .), 36 (36, 72, 108, 144, 180 . . .), and 45 (45, 90, 135, 180, 225 . . .).
Step 2: The common multiples from the multiples of 28, 36, and 45 are 1260, 2520, . . .
Step 3: The smallest common multiple of 28, 36, and 45 is 1260.

∴ The least common multiple of 28, 36, and 45 = 1260.

LCM of 28, 36, and 45 by Division Method

To calculate the LCM of 28, 36, and 45 by the division method, we will divide the numbers(28, 36, 45) by their prime factors (preferably common). The product of these divisors gives the LCM of 28, 36, and 45.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 28, 36, and 45. Write this prime number(2) on the left of the given numbers(28, 36, and 45), separated as per the ladder arrangement.
Step 2: If any of the given numbers (28, 36, 45) 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, 36, and 45 is the product of all prime numbers on the left, i.e. LCM(28, 36, 45) by division method = 2 × 2 × 3 × 3 × 5 × 7 = 1260.

LCM of 28, 36, and 45 by Prime Factorization

Prime factorization of 28, 36, and 45 is (2 × 2 × 7) = 2² × 7¹, (2 × 2 × 3 × 3) = 2² × 3², and (3 × 3 × 5) = 3² × 5¹ respectively. LCM of 28, 36, and 45 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2² × 3² × 5¹ × 7¹ = 1260.
Hence, the LCM of 28, 36, and 45 by prime factorization is 1260.

☛ Also Check:

LCM of 28, 36, and 45 Examples

Example 1: Calculate the LCM of 28, 36, and 45 using the GCD of the given numbers.

Solution:

Prime factorization of 28, 36, 45:
- 28 = 2² × 7¹
- 36 = 2² × 3²
- 45 = 3² × 5¹
Therefore, GCD(28, 36) = 4, GCD(36, 45) = 9, GCD(28, 45) = 1, GCD(28, 36, 45) = 1
We know,
LCM(28, 36, 45) = [(28 × 36 × 45) × GCD(28, 36, 45)]/[GCD(28, 36) × GCD(36, 45) × GCD(28, 45)]
LCM(28, 36, 45) = (45360 × 1)/(4 × 9 × 1) = 1260
⇒LCM(28, 36, 45) = 1260
Example 2: Verify the relationship between the GCD and LCM of 28, 36, and 45.

Solution:

The relation between GCD and LCM of 28, 36, and 45 is given as,
LCM(28, 36, 45) = [(28 × 36 × 45) × GCD(28, 36, 45)]/[GCD(28, 36) × GCD(36, 45) × GCD(28, 45)]
⇒ Prime factorization of 28, 36 and 45:
- 28 = 2² × 7¹
- 36 = 2² × 3²
- 45 = 3² × 5¹
∴ GCD of (28, 36), (36, 45), (28, 45) and (28, 36, 45) = 4, 9, 1 and 1 respectively.
Now, LHS = LCM(28, 36, 45) = 1260.
And, RHS = [(28 × 36 × 45) × GCD(28, 36, 45)]/[GCD(28, 36) × GCD(36, 45) × GCD(28, 45)] = [(45360) × 1]/[4 × 9 × 1] = 1260
LHS = RHS = 1260.
Hence verified.
Example 3: Find the smallest number that is divisible by 28, 36, 45 exactly.

Solution:

The value of LCM(28, 36, 45) will be the smallest number that is exactly divisible by 28, 36, and 45.
⇒ Multiples of 28, 36, and 45:
- Multiples of 28 = 28, 56, 84, 112, 140, 168, 196, 224, 252, 280, . . . ., 1204, 1232, 1260, . . . .
- Multiples of 36 = 36, 72, 108, 144, 180, 216, 252, 288, 324, 360, . . . ., 1152, 1188, 1224, 1260, . . . .
- Multiples of 45 = 45, 90, 135, 180, 225, 270, 315, 360, 405, 450, . . . ., 1170, 1215, 1260, . . . .
Therefore, the LCM of 28, 36, and 45 is 1260.

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FAQs on LCM of 28, 36, and 45

What is the LCM of 28, 36, and 45?

The LCM of 28, 36, and 45 is 1260. To find the least common multiple of 28, 36, and 45, we need to find the multiples of 28, 36, and 45 (multiples of 28 = 28, 56, 84, 112 . . . . 1260 . . . . ; multiples of 36 = 36, 72, 108, 144 . . . . 1260 . . . . ; multiples of 45 = 45, 90, 135, 180 . . . . 1260 . . . . ) and choose the smallest multiple that is exactly divisible by 28, 36, and 45, i.e., 1260.

How to Find the LCM of 28, 36, and 45 by Prime Factorization?

To find the LCM of 28, 36, and 45 using prime factorization, we will find the prime factors, (28 = 2² × 7¹), (36 = 2² × 3²), and (45 = 3² × 5¹). LCM of 28, 36, and 45 is the product of prime factors raised to their respective highest exponent among the numbers 28, 36, and 45.
⇒ LCM of 28, 36, 45 = 2² × 3² × 5¹ × 7¹ = 1260.

What is the Relation Between GCF and LCM of 28, 36, 45?

The following equation can be used to express the relation between GCF and LCM of 28, 36, 45, i.e. LCM(28, 36, 45) = [(28 × 36 × 45) × GCF(28, 36, 45)]/[GCF(28, 36) × GCF(36, 45) × GCF(28, 45)].

Which of the following is the LCM of 28, 36, and 45? 1260, 40, 100, 12

The value of LCM of 28, 36, 45 is the smallest common multiple of 28, 36, and 45. The number satisfying the given condition is 1260.

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