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LCM of 75 and 100

LCM of 75 and 100 is the smallest number among all common multiples of 75 and 100. The first few multiples of 75 and 100 are (75, 150, 225, 300, 375, . . . ) and (100, 200, 300, 400, 500, 600, 700, . . . ) respectively. There are 3 commonly used methods to find LCM of 75 and 100 - by prime factorization, by listing multiples, and by division method.

1.	LCM of 75 and 100
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 75 and 100?

Answer: LCM of 75 and 100 is 300.

Explanation:

The LCM of two non-zero integers, x(75) and y(100), is the smallest positive integer m(300) that is divisible by both x(75) and y(100) without any remainder.

Methods to Find LCM of 75 and 100

The methods to find the LCM of 75 and 100 are explained below.

By Prime Factorization Method
By Division Method
By Listing Multiples

LCM of 75 and 100 by Prime Factorization

Prime factorization of 75 and 100 is (3 × 5 × 5) = 3¹ × 5² and (2 × 2 × 5 × 5) = 2² × 5² respectively. LCM of 75 and 100 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2² × 3¹ × 5² = 300.
Hence, the LCM of 75 and 100 by prime factorization is 300.

LCM of 75 and 100 by Division Method

To calculate the LCM of 75 and 100 by the division method, we will divide the numbers(75, 100) by their prime factors (preferably common). The product of these divisors gives the LCM of 75 and 100.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 75 and 100. Write this prime number(2) on the left of the given numbers(75 and 100), separated as per the ladder arrangement.
Step 2: If any of the given numbers (75, 100) 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 75 and 100 is the product of all prime numbers on the left, i.e. LCM(75, 100) by division method = 2 × 2 × 3 × 5 × 5 = 300.

LCM of 75 and 100 by Listing Multiples

To calculate the LCM of 75 and 100 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 75 (75, 150, 225, 300, 375, . . . ) and 100 (100, 200, 300, 400, 500, 600, 700, . . . . )
Step 2: The common multiples from the multiples of 75 and 100 are 300, 600, . . .
Step 3: The smallest common multiple of 75 and 100 is 300.

∴ The least common multiple of 75 and 100 = 300.

☛ Also Check:

LCM of 75 and 100 Examples

Example 1: Verify the relationship between GCF and LCM of 75 and 100.

Solution:

The relation between GCF and LCM of 75 and 100 is given as,
LCM(75, 100) × GCF(75, 100) = Product of 75, 100
Prime factorization of 75 and 100 is given as, 75 = (3 × 5 × 5) = 3¹ × 5² and 100 = (2 × 2 × 5 × 5) = 2² × 5²
LCM(75, 100) = 300
GCF(75, 100) = 25
LHS = LCM(75, 100) × GCF(75, 100) = 300 × 25 = 7500
RHS = Product of 75, 100 = 75 × 100 = 7500
⇒ LHS = RHS = 7500
Hence, verified.
Example 2: The product of two numbers is 7500. If their GCD is 25, what is their LCM?

Solution:

Given: GCD = 25
product of numbers = 7500
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 7500/25
Therefore, the LCM is 300.
The probable combination for the given case is LCM(75, 100) = 300.
Example 3: The GCD and LCM of two numbers are 25 and 300 respectively. If one number is 75, find the other number.

Solution:

Let the other number be a.
∵ GCD × LCM = 75 × a
⇒ a = (GCD × LCM)/75
⇒ a = (25 × 300)/75
⇒ a = 100
Therefore, the other number is 100.

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FAQs on LCM of 75 and 100

What is the LCM of 75 and 100?

The LCM of 75 and 100 is 300. To find the least common multiple (LCM) of 75 and 100, we need to find the multiples of 75 and 100 (multiples of 75 = 75, 150, 225, 300; multiples of 100 = 100, 200, 300, 400) and choose the smallest multiple that is exactly divisible by 75 and 100, i.e., 300.

How to Find the LCM of 75 and 100 by Prime Factorization?

To find the LCM of 75 and 100 using prime factorization, we will find the prime factors, (75 = 3 × 5 × 5) and (100 = 2 × 2 × 5 × 5). LCM of 75 and 100 is the product of prime factors raised to their respective highest exponent among the numbers 75 and 100.
⇒ LCM of 75, 100 = 2² × 3¹ × 5² = 300.

If the LCM of 100 and 75 is 300, Find its GCF.

LCM(100, 75) × GCF(100, 75) = 100 × 75
Since the LCM of 100 and 75 = 300
⇒ 300 × GCF(100, 75) = 7500
Therefore, the GCF (greatest common factor) = 7500/300 = 25.

What is the Relation Between GCF and LCM of 75, 100?

The following equation can be used to express the relation between GCF and LCM of 75 and 100, i.e. GCF × LCM = 75 × 100.

What is the Least Perfect Square Divisible by 75 and 100?

The least number divisible by 75 and 100 = LCM(75, 100)
LCM of 75 and 100 = 2 × 2 × 3 × 5 × 5 [Incomplete pair(s): 3]
⇒ Least perfect square divisible by each 75 and 100 = LCM(75, 100) × 3 = 900 [Square root of 900 = √900 = ±30]
Therefore, 900 is the required number.

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