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LCM of 40 and 45

LCM of 40 and 45 is the smallest number among all common multiples of 40 and 45. The first few multiples of 40 and 45 are (40, 80, 120, 160, 200, 240, 280, . . . ) and (45, 90, 135, 180, 225, . . . ) respectively. There are 3 commonly used methods to find LCM of 40 and 45 - by prime factorization, by division method, and by listing multiples.

1.	LCM of 40 and 45
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 40 and 45?

Answer: LCM of 40 and 45 is 360.

Explanation:

The LCM of two non-zero integers, x(40) and y(45), is the smallest positive integer m(360) that is divisible by both x(40) and y(45) without any remainder.

Methods to Find LCM of 40 and 45

Let's look at the different methods for finding the LCM of 40 and 45.

By Prime Factorization Method
By Listing Multiples
By Division Method

LCM of 40 and 45 by Prime Factorization

Prime factorization of 40 and 45 is (2 × 2 × 2 × 5) = 2³ × 5¹ and (3 × 3 × 5) = 3² × 5¹ respectively. LCM of 40 and 45 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2³ × 3² × 5¹ = 360.
Hence, the LCM of 40 and 45 by prime factorization is 360.

LCM of 40 and 45 by Listing Multiples

To calculate the LCM of 40 and 45 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 40 (40, 80, 120, 160, 200, 240, 280, . . . ) and 45 (45, 90, 135, 180, 225, . . . . )
Step 2: The common multiples from the multiples of 40 and 45 are 360, 720, . . .
Step 3: The smallest common multiple of 40 and 45 is 360.

∴ The least common multiple of 40 and 45 = 360.

LCM of 40 and 45 by Division Method

To calculate the LCM of 40 and 45 by the division method, we will divide the numbers(40, 45) by their prime factors (preferably common). The product of these divisors gives the LCM of 40 and 45.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 40 and 45. Write this prime number(2) on the left of the given numbers(40 and 45), separated as per the ladder arrangement.
Step 2: If any of the given numbers (40, 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 40 and 45 is the product of all prime numbers on the left, i.e. LCM(40, 45) by division method = 2 × 2 × 2 × 3 × 3 × 5 = 360.

☛ Also Check:

LCM of 40 and 45 Examples

Example 1: The product of two numbers is 1800. If their GCD is 5, what is their LCM?

Solution:

Given: GCD = 5
product of numbers = 1800
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 1800/5
Therefore, the LCM is 360.
The probable combination for the given case is LCM(40, 45) = 360.
Example 2: The GCD and LCM of two numbers are 5 and 360 respectively. If one number is 40, find the other number.

Solution:

Let the other number be a.
∵ GCD × LCM = 40 × a
⇒ a = (GCD × LCM)/40
⇒ a = (5 × 360)/40
⇒ a = 45
Therefore, the other number is 45.
Example 3: Verify the relationship between GCF and LCM of 40 and 45.

Solution:

The relation between GCF and LCM of 40 and 45 is given as,
LCM(40, 45) × GCF(40, 45) = Product of 40, 45
Prime factorization of 40 and 45 is given as, 40 = (2 × 2 × 2 × 5) = 2³ × 5¹ and 45 = (3 × 3 × 5) = 3² × 5¹
LCM(40, 45) = 360
GCF(40, 45) = 5
LHS = LCM(40, 45) × GCF(40, 45) = 360 × 5 = 1800
RHS = Product of 40, 45 = 40 × 45 = 1800
⇒ LHS = RHS = 1800
Hence, verified.

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FAQs on LCM of 40 and 45

What is the LCM of 40 and 45?

The LCM of 40 and 45 is 360. To find the LCM (least common multiple) of 40 and 45, we need to find the multiples of 40 and 45 (multiples of 40 = 40, 80, 120, 160 . . . . 360; multiples of 45 = 45, 90, 135, 180 . . . . 360) and choose the smallest multiple that is exactly divisible by 40 and 45, i.e., 360.

If the LCM of 45 and 40 is 360, Find its GCF.

LCM(45, 40) × GCF(45, 40) = 45 × 40
Since the LCM of 45 and 40 = 360
⇒ 360 × GCF(45, 40) = 1800
Therefore, the greatest common factor (GCF) = 1800/360 = 5.

What are the Methods to Find LCM of 40 and 45?

The commonly used methods to find the LCM of 40 and 45 are:

Listing Multiples
Division Method
Prime Factorization Method

How to Find the LCM of 40 and 45 by Prime Factorization?

To find the LCM of 40 and 45 using prime factorization, we will find the prime factors, (40 = 2 × 2 × 2 × 5) and (45 = 3 × 3 × 5). LCM of 40 and 45 is the product of prime factors raised to their respective highest exponent among the numbers 40 and 45.
⇒ LCM of 40, 45 = 2³ × 3² × 5¹ = 360.

Which of the following is the LCM of 40 and 45? 10, 45, 360, 5

The value of LCM of 40, 45 is the smallest common multiple of 40 and 45. The number satisfying the given condition is 360.

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