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LCM of 12 and 25

LCM of 12 and 25 is the smallest number among all common multiples of 12 and 25. The first few multiples of 12 and 25 are (12, 24, 36, 48, 60, 72, . . . ) and (25, 50, 75, 100, 125, 150, 175, . . . ) respectively. There are 3 commonly used methods to find LCM of 12 and 25 - by listing multiples, by prime factorization, and by division method.

1.	LCM of 12 and 25
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 12 and 25?

Answer: LCM of 12 and 25 is 300.

Explanation:

The LCM of two non-zero integers, x(12) and y(25), is the smallest positive integer m(300) that is divisible by both x(12) and y(25) without any remainder.

Methods to Find LCM of 12 and 25

Let's look at the different methods for finding the LCM of 12 and 25.

By Listing Multiples
By Prime Factorization Method
By Division Method

LCM of 12 and 25 by Listing Multiples

To calculate the LCM of 12 and 25 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 12 (12, 24, 36, 48, 60, 72, . . . ) and 25 (25, 50, 75, 100, 125, 150, 175, . . . . )
Step 2: The common multiples from the multiples of 12 and 25 are 300, 600, . . .
Step 3: The smallest common multiple of 12 and 25 is 300.

∴ The least common multiple of 12 and 25 = 300.

LCM of 12 and 25 by Prime Factorization

Prime factorization of 12 and 25 is (2 × 2 × 3) = 2² × 3¹ and (5 × 5) = 5² respectively. LCM of 12 and 25 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2² × 3¹ × 5² = 300.
Hence, the LCM of 12 and 25 by prime factorization is 300.

LCM of 12 and 25 by Division Method

To calculate the LCM of 12 and 25 by the division method, we will divide the numbers(12, 25) by their prime factors (preferably common). The product of these divisors gives the LCM of 12 and 25.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 12 and 25. Write this prime number(2) on the left of the given numbers(12 and 25), separated as per the ladder arrangement.
Step 2: If any of the given numbers (12, 25) 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 12 and 25 is the product of all prime numbers on the left, i.e. LCM(12, 25) by division method = 2 × 2 × 3 × 5 × 5 = 300.

☛ Also Check:

LCM of 12 and 25 Examples

Example 1: Find the smallest number that is divisible by 12 and 25 exactly.

Solution:

The value of LCM(12, 25) will be the smallest number that is exactly divisible by 12 and 25.
⇒ Multiples of 12 and 25:
- Multiples of 12 = 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, . . . ., 264, 276, 288, 300, . . . .
- Multiples of 25 = 25, 50, 75, 100, 125, 150, 175, 200, 225, 250, . . . ., 225, 250, 275, 300, . . . .
Therefore, the LCM of 12 and 25 is 300.
Example 2: Verify the relationship between GCF and LCM of 12 and 25.

Solution:

The relation between GCF and LCM of 12 and 25 is given as,
LCM(12, 25) × GCF(12, 25) = Product of 12, 25
Prime factorization of 12 and 25 is given as, 12 = (2 × 2 × 3) = 2² × 3¹ and 25 = (5 × 5) = 5²
LCM(12, 25) = 300
GCF(12, 25) = 1
LHS = LCM(12, 25) × GCF(12, 25) = 300 × 1 = 300
RHS = Product of 12, 25 = 12 × 25 = 300
⇒ LHS = RHS = 300
Hence, verified.
Example 3: The GCD and LCM of two numbers are 1 and 300 respectively. If one number is 25, find the other number.

Solution:

Let the other number be m.
∵ GCD × LCM = 25 × m
⇒ m = (GCD × LCM)/25
⇒ m = (1 × 300)/25
⇒ m = 12
Therefore, the other number is 12.

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FAQs on LCM of 12 and 25

What is the LCM of 12 and 25?

The LCM of 12 and 25 is 300. To find the least common multiple of 12 and 25, we need to find the multiples of 12 and 25 (multiples of 12 = 12, 24, 36, 48 . . . . 300; multiples of 25 = 25, 50, 75, 100 . . . . 300) and choose the smallest multiple that is exactly divisible by 12 and 25, i.e., 300.

What is the Least Perfect Square Divisible by 12 and 25?

The least number divisible by 12 and 25 = LCM(12, 25)
LCM of 12 and 25 = 2 × 2 × 3 × 5 × 5 [Incomplete pair(s): 3]
⇒ Least perfect square divisible by each 12 and 25 = LCM(12, 25) × 3 = 900 [Square root of 900 = √900 = ±30]
Therefore, 900 is the required number.

How to Find the LCM of 12 and 25 by Prime Factorization?

To find the LCM of 12 and 25 using prime factorization, we will find the prime factors, (12 = 2 × 2 × 3) and (25 = 5 × 5). LCM of 12 and 25 is the product of prime factors raised to their respective highest exponent among the numbers 12 and 25.
⇒ LCM of 12, 25 = 2² × 3¹ × 5² = 300.

If the LCM of 25 and 12 is 300, Find its GCF.

LCM(25, 12) × GCF(25, 12) = 25 × 12
Since the LCM of 25 and 12 = 300
⇒ 300 × GCF(25, 12) = 300
Therefore, the GCF (greatest common factor) = 300/300 = 1.

What is the Relation Between GCF and LCM of 12, 25?

The following equation can be used to express the relation between GCF and LCM of 12 and 25, i.e. GCF × LCM = 12 × 25.

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