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LCM of 10, 15, and 25

LCM of 10, 15, and 25 is the smallest number among all common multiples of 10, 15, and 25. The first few multiples of 10, 15, and 25 are (10, 20, 30, 40, 50 . . .), (15, 30, 45, 60, 75 . . .), and (25, 50, 75, 100, 125 . . .) respectively. There are 3 commonly used methods to find LCM of 10, 15, 25 - by prime factorization, by division method, and by listing multiples.

1.	LCM of 10, 15, and 25
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 10, 15, and 25?

Answer: LCM of 10, 15, and 25 is 150.

Explanation:

The LCM of three non-zero integers, a(10), b(15), and c(25), is the smallest positive integer m(150) that is divisible by a(10), b(15), and c(25) without any remainder.

Methods to Find LCM of 10, 15, and 25

The methods to find the LCM of 10, 15, and 25 are explained below.

By Prime Factorization Method
By Listing Multiples
By Division Method

LCM of 10, 15, and 25 by Prime Factorization

Prime factorization of 10, 15, and 25 is (2 × 5) = 2¹ × 5¹, (3 × 5) = 3¹ × 5¹, and (5 × 5) = 5² respectively. LCM of 10, 15, and 25 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2¹ × 3¹ × 5² = 150.
Hence, the LCM of 10, 15, and 25 by prime factorization is 150.

LCM of 10, 15, and 25 by Listing Multiples

To calculate the LCM of 10, 15, 25 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 10 (10, 20, 30, 40, 50 . . .), 15 (15, 30, 45, 60, 75 . . .), and 25 (25, 50, 75, 100, 125 . . .).
Step 2: The common multiples from the multiples of 10, 15, and 25 are 150, 300, . . .
Step 3: The smallest common multiple of 10, 15, and 25 is 150.

∴ The least common multiple of 10, 15, and 25 = 150.

LCM of 10, 15, and 25 by Division Method

To calculate the LCM of 10, 15, and 25 by the division method, we will divide the numbers(10, 15, 25) by their prime factors (preferably common). The product of these divisors gives the LCM of 10, 15, and 25.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 10, 15, and 25. Write this prime number(2) on the left of the given numbers(10, 15, and 25), separated as per the ladder arrangement.
Step 2: If any of the given numbers (10, 15, 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 10, 15, and 25 is the product of all prime numbers on the left, i.e. LCM(10, 15, 25) by division method = 2 × 3 × 5 × 5 = 150.

☛ Also Check:

LCM of 10, 15, and 25 Examples

Example 1: Find the smallest number that is divisible by 10, 15, 25 exactly.

Solution:

The smallest number that is divisible by 10, 15, and 25 exactly is their LCM.
⇒ Multiples of 10, 15, and 25:
- Multiples of 10 = 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 130, 140, 150, . . . .
- Multiples of 15 = 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, . . . .
- Multiples of 25 = 25, 50, 75, 100, 125, 150, . . . .
Therefore, the LCM of 10, 15, and 25 is 150.
Example 2: Verify the relationship between the GCD and LCM of 10, 15, and 25.

Solution:

The relation between GCD and LCM of 10, 15, and 25 is given as,
LCM(10, 15, 25) = [(10 × 15 × 25) × GCD(10, 15, 25)]/[GCD(10, 15) × GCD(15, 25) × GCD(10, 25)]
⇒ Prime factorization of 10, 15 and 25:
- 10 = 2¹ × 5¹
- 15 = 3¹ × 5¹
- 25 = 5²
∴ GCD of (10, 15), (15, 25), (10, 25) and (10, 15, 25) = 5, 5, 5 and 5 respectively.
Now, LHS = LCM(10, 15, 25) = 150.
And, RHS = [(10 × 15 × 25) × GCD(10, 15, 25)]/[GCD(10, 15) × GCD(15, 25) × GCD(10, 25)] = [(3750) × 5]/[5 × 5 × 5] = 150
LHS = RHS = 150.
Hence verified.
Example 3: Calculate the LCM of 10, 15, and 25 using the GCD of the given numbers.

Solution:

Prime factorization of 10, 15, 25:
- 10 = 2¹ × 5¹
- 15 = 3¹ × 5¹
- 25 = 5²
Therefore, GCD(10, 15) = 5, GCD(15, 25) = 5, GCD(10, 25) = 5, GCD(10, 15, 25) = 5
We know,
LCM(10, 15, 25) = [(10 × 15 × 25) × GCD(10, 15, 25)]/[GCD(10, 15) × GCD(15, 25) × GCD(10, 25)]
LCM(10, 15, 25) = (3750 × 5)/(5 × 5 × 5) = 150
⇒LCM(10, 15, 25) = 150

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FAQs on LCM of 10, 15, and 25

What is the LCM of 10, 15, and 25?

The LCM of 10, 15, and 25 is 150. To find the least common multiple (LCM) of 10, 15, and 25, we need to find the multiples of 10, 15, and 25 (multiples of 10 = 10, 20, 30, 40 . . . . 150 . . . . ; multiples of 15 = 15, 30, 45, 60 . . . . 150 . . . . ; multiples of 25 = 25, 50, 75, 100, 150 . . . .) and choose the smallest multiple that is exactly divisible by 10, 15, and 25, i.e., 150.

Which of the following is the LCM of 10, 15, and 25? 25, 150, 35, 10

The value of LCM of 10, 15, 25 is the smallest common multiple of 10, 15, and 25. The number satisfying the given condition is 150.

How to Find the LCM of 10, 15, and 25 by Prime Factorization?

To find the LCM of 10, 15, and 25 using prime factorization, we will find the prime factors, (10 = 2¹ × 5¹), (15 = 3¹ × 5¹), and (25 = 5²). LCM of 10, 15, and 25 is the product of prime factors raised to their respective highest exponent among the numbers 10, 15, and 25.
⇒ LCM of 10, 15, 25 = 2¹ × 3¹ × 5² = 150.

What is the Least Perfect Square Divisible by 10, 15, and 25?

The least number divisible by 10, 15, and 25 = LCM(10, 15, 25)
LCM of 10, 15, and 25 = 2 × 3 × 5 × 5 [Incomplete pair(s): 2, 3]
⇒ Least perfect square divisible by each 10, 15, and 25 = LCM(10, 15, 25) × 2 × 3 = 900 [Square root of 900 = √900 = ±30]
Therefore, 900 is the required number.

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