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LCM of 45 and 60

LCM of 45 and 60 is the smallest number among all common multiples of 45 and 60. The first few multiples of 45 and 60 are (45, 90, 135, 180, 225, 270, . . . ) and (60, 120, 180, 240, 300, 360, . . . ) respectively. There are 3 commonly used methods to find LCM of 45 and 60 - by listing multiples, by prime factorization, and by division method.

1.	LCM of 45 and 60
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 45 and 60?

Answer: LCM of 45 and 60 is 180.

Explanation:

The LCM of two non-zero integers, x(45) and y(60), is the smallest positive integer m(180) that is divisible by both x(45) and y(60) without any remainder.

Methods to Find LCM of 45 and 60

Let's look at the different methods for finding the LCM of 45 and 60.

By Listing Multiples
By Prime Factorization Method
By Division Method

LCM of 45 and 60 by Listing Multiples

To calculate the LCM of 45 and 60 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 45 (45, 90, 135, 180, 225, 270, . . . ) and 60 (60, 120, 180, 240, 300, 360, . . . . )
Step 2: The common multiples from the multiples of 45 and 60 are 180, 360, . . .
Step 3: The smallest common multiple of 45 and 60 is 180.

∴ The least common multiple of 45 and 60 = 180.

LCM of 45 and 60 by Prime Factorization

Prime factorization of 45 and 60 is (3 × 3 × 5) = 3² × 5¹ and (2 × 2 × 3 × 5) = 2² × 3¹ × 5¹ respectively. LCM of 45 and 60 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2² × 3² × 5¹ = 180.
Hence, the LCM of 45 and 60 by prime factorization is 180.

LCM of 45 and 60 by Division Method

To calculate the LCM of 45 and 60 by the division method, we will divide the numbers(45, 60) by their prime factors (preferably common). The product of these divisors gives the LCM of 45 and 60.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 45 and 60. Write this prime number(2) on the left of the given numbers(45 and 60), separated as per the ladder arrangement.
Step 2: If any of the given numbers (45, 60) 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 45 and 60 is the product of all prime numbers on the left, i.e. LCM(45, 60) by division method = 2 × 2 × 3 × 3 × 5 = 180.

☛ Also Check:

LCM of 45 and 60 Examples

Example 1: Find the smallest number that is divisible by 45 and 60 exactly.

Solution:

The smallest number that is divisible by 45 and 60 exactly is their LCM.
⇒ Multiples of 45 and 60:
- Multiples of 45 = 45, 90, 135, 180, 225, . . . .
- Multiples of 60 = 60, 120, 180, 240, 300, . . . .
Therefore, the LCM of 45 and 60 is 180.
Example 2: Verify the relationship between GCF and LCM of 45 and 60.

Solution:

The relation between GCF and LCM of 45 and 60 is given as,
LCM(45, 60) × GCF(45, 60) = Product of 45, 60
Prime factorization of 45 and 60 is given as, 45 = (3 × 3 × 5) = 3² × 5¹ and 60 = (2 × 2 × 3 × 5) = 2² × 3¹ × 5¹
LCM(45, 60) = 180
GCF(45, 60) = 15
LHS = LCM(45, 60) × GCF(45, 60) = 180 × 15 = 2700
RHS = Product of 45, 60 = 45 × 60 = 2700
⇒ LHS = RHS = 2700
Hence, verified.
Example 3: The product of two numbers is 2700. If their GCD is 15, what is their LCM?

Solution:

Given: GCD = 15
product of numbers = 2700
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 2700/15
Therefore, the LCM is 180.
The probable combination for the given case is LCM(45, 60) = 180.

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FAQs on LCM of 45 and 60

What is the LCM of 45 and 60?

The LCM of 45 and 60 is 180. To find the least common multiple of 45 and 60, we need to find the multiples of 45 and 60 (multiples of 45 = 45, 90, 135, 180; multiples of 60 = 60, 120, 180, 240) and choose the smallest multiple that is exactly divisible by 45 and 60, i.e., 180.

Which of the following is the LCM of 45 and 60? 11, 35, 16, 180

The value of LCM of 45, 60 is the smallest common multiple of 45 and 60. The number satisfying the given condition is 180.

What is the Least Perfect Square Divisible by 45 and 60?

The least number divisible by 45 and 60 = LCM(45, 60)
LCM of 45 and 60 = 2 × 2 × 3 × 3 × 5 [Incomplete pair(s): 5]
⇒ Least perfect square divisible by each 45 and 60 = LCM(45, 60) × 5 = 900 [Square root of 900 = √900 = ±30]
Therefore, 900 is the required number.

What is the Relation Between GCF and LCM of 45, 60?

The following equation can be used to express the relation between GCF and LCM of 45 and 60, i.e. GCF × LCM = 45 × 60.

If the LCM of 60 and 45 is 180, Find its GCF.

LCM(60, 45) × GCF(60, 45) = 60 × 45
Since the LCM of 60 and 45 = 180
⇒ 180 × GCF(60, 45) = 2700
Therefore, the GCF = 2700/180 = 15.

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