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LCM of 45, 60, and 75

LCM of 45, 60, and 75 is the smallest number among all common multiples of 45, 60, and 75. The first few multiples of 45, 60, and 75 are (45, 90, 135, 180, 225 . . .), (60, 120, 180, 240, 300 . . .), and (75, 150, 225, 300, 375 . . .) respectively. There are 3 commonly used methods to find LCM of 45, 60, 75 - by listing multiples, by division method, and by prime factorization.

1.	LCM of 45, 60, and 75
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 45, 60, and 75?

Answer: LCM of 45, 60, and 75 is 900.

Explanation:

The LCM of three non-zero integers, a(45), b(60), and c(75), is the smallest positive integer m(900) that is divisible by a(45), b(60), and c(75) without any remainder.

Methods to Find LCM of 45, 60, and 75

The methods to find the LCM of 45, 60, and 75 are explained below.

By Listing Multiples
By Division Method
By Prime Factorization Method

LCM of 45, 60, and 75 by Listing Multiples

To calculate the LCM of 45, 60, 75 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 . . .), 60 (60, 120, 180, 240, 300 . . .), and 75 (75, 150, 225, 300, 375 . . .).
Step 2: The common multiples from the multiples of 45, 60, and 75 are 900, 1800, . . .
Step 3: The smallest common multiple of 45, 60, and 75 is 900.

∴ The least common multiple of 45, 60, and 75 = 900.

LCM of 45, 60, and 75 by Division Method

To calculate the LCM of 45, 60, and 75 by the division method, we will divide the numbers(45, 60, 75) by their prime factors (preferably common). The product of these divisors gives the LCM of 45, 60, and 75.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 45, 60, and 75. Write this prime number(2) on the left of the given numbers(45, 60, and 75), separated as per the ladder arrangement.
Step 2: If any of the given numbers (45, 60, 75) 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, 60, and 75 is the product of all prime numbers on the left, i.e. LCM(45, 60, 75) by division method = 2 × 2 × 3 × 3 × 5 × 5 = 900.

LCM of 45, 60, and 75 by Prime Factorization

Prime factorization of 45, 60, and 75 is (3 × 3 × 5) = 3² × 5¹, (2 × 2 × 3 × 5) = 2² × 3¹ × 5¹, and (3 × 5 × 5) = 3¹ × 5² respectively. LCM of 45, 60, and 75 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2² × 3² × 5² = 900.
Hence, the LCM of 45, 60, and 75 by prime factorization is 900.

☛ Also Check:

LCM of 45, 60, and 75 Examples

Example 1: Verify the relationship between the GCD and LCM of 45, 60, and 75.

Solution:

The relation between GCD and LCM of 45, 60, and 75 is given as,
LCM(45, 60, 75) = [(45 × 60 × 75) × GCD(45, 60, 75)]/[GCD(45, 60) × GCD(60, 75) × GCD(45, 75)]
⇒ Prime factorization of 45, 60 and 75:
- 45 = 3² × 5¹
- 60 = 2² × 3¹ × 5¹
- 75 = 3¹ × 5²
∴ GCD of (45, 60), (60, 75), (45, 75) and (45, 60, 75) = 15, 15, 15 and 15 respectively.
Now, LHS = LCM(45, 60, 75) = 900.
And, RHS = [(45 × 60 × 75) × GCD(45, 60, 75)]/[GCD(45, 60) × GCD(60, 75) × GCD(45, 75)] = [(202500) × 15]/[15 × 15 × 15] = 900
LHS = RHS = 900.
Hence verified.
Example 2: Calculate the LCM of 45, 60, and 75 using the GCD of the given numbers.

Solution:

Prime factorization of 45, 60, 75:
- 45 = 3² × 5¹
- 60 = 2² × 3¹ × 5¹
- 75 = 3¹ × 5²
Therefore, GCD(45, 60) = 15, GCD(60, 75) = 15, GCD(45, 75) = 15, GCD(45, 60, 75) = 15
We know,
LCM(45, 60, 75) = [(45 × 60 × 75) × GCD(45, 60, 75)]/[GCD(45, 60) × GCD(60, 75) × GCD(45, 75)]
LCM(45, 60, 75) = (202500 × 15)/(15 × 15 × 15) = 900
⇒LCM(45, 60, 75) = 900
Example 3: Find the smallest number that is divisible by 45, 60, 75 exactly.

Solution:

The value of LCM(45, 60, 75) will be the smallest number that is exactly divisible by 45, 60, and 75.
⇒ Multiples of 45, 60, and 75:
- Multiples of 45 = 45, 90, 135, 180, 225, 270, 315, 360, 405, 450, . . . ., 765, 810, 855, 900, . . . .
- Multiples of 60 = 60, 120, 180, 240, 300, 360, 420, 480, 540, 600, . . . ., 720, 780, 840, 900, . . . .
- Multiples of 75 = 75, 150, 225, 300, 375, 450, 525, 600, 675, 750, . . . ., 600, 675, 750, 825, 900, . . . .
Therefore, the LCM of 45, 60, and 75 is 900.

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FAQs on LCM of 45, 60, and 75

What is the LCM of 45, 60, and 75?

The LCM of 45, 60, and 75 is 900. To find the LCM (least common multiple) of 45, 60, and 75, we need to find the multiples of 45, 60, and 75 (multiples of 45 = 45, 90, 135, 180 . . . . 900 . . . . ; multiples of 60 = 60, 120, 180, 240 . . . . 900 . . . . ; multiples of 75 = 75, 150, 225, 300 . . . . 900 . . . . ) and choose the smallest multiple that is exactly divisible by 45, 60, and 75, i.e., 900.

What is the Least Perfect Square Divisible by 45, 60, and 75?

The least number divisible by 45, 60, and 75 = LCM(45, 60, 75)
LCM of 45, 60, and 75 = 2 × 2 × 3 × 3 × 5 × 5 [No incomplete pair]
⇒ Least perfect square divisible by each 45, 60, and 75 = LCM(45, 60, 75) = 900 [Square root of 900 = √900 = ±30]
Therefore, 900 is the required number.

What are the Methods to Find LCM of 45, 60, 75?

The commonly used methods to find the LCM of 45, 60, 75 are:

Listing Multiples
Prime Factorization Method
Division Method

What is the Relation Between GCF and LCM of 45, 60, 75?

The following equation can be used to express the relation between GCF and LCM of 45, 60, 75, i.e. LCM(45, 60, 75) = [(45 × 60 × 75) × GCF(45, 60, 75)]/[GCF(45, 60) × GCF(60, 75) × GCF(45, 75)].

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