LCM of 15, 25, 40, and 75
LCM of 15, 25, 40, and 75 is the smallest number among all common multiples of 15, 25, 40, and 75. The first few multiples of 15, 25, 40, and 75 are (15, 30, 45, 60, 75 . . .), (25, 50, 75, 100, 125 . . .), (40, 80, 120, 160, 200 . . .), and (75, 150, 225, 300, 375 . . .) respectively. There are 3 commonly used methods to find LCM of 15, 25, 40, 75  by listing multiples, by division method, and by prime factorization.
1.  LCM of 15, 25, 40, and 75 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is the LCM of 15, 25, 40, and 75?
Answer: LCM of 15, 25, 40, and 75 is 600.
Explanation:
The LCM of four nonzero integers, a(15), b(25), c(40), and d(75), is the smallest positive integer m(600) that is divisible by a(15), b(25), c(40), and d(75) without any remainder.
Methods to Find LCM of 15, 25, 40, and 75
Let's look at the different methods for finding the LCM of 15, 25, 40, and 75.
 By Division Method
 By Prime Factorization Method
 By Listing Multiples
LCM of 15, 25, 40, and 75 by Division Method
To calculate the LCM of 15, 25, 40, and 75 by the division method, we will divide the numbers(15, 25, 40, 75) by their prime factors (preferably common). The product of these divisors gives the LCM of 15, 25, 40, and 75.
 Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 15, 25, 40, and 75. Write this prime number(2) on the left of the given numbers(15, 25, 40, and 75), separated as per the ladder arrangement.
 Step 2: If any of the given numbers (15, 25, 40, 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 15, 25, 40, and 75 is the product of all prime numbers on the left, i.e. LCM(15, 25, 40, 75) by division method = 2 × 2 × 2 × 3 × 5 × 5 = 600.
LCM of 15, 25, 40, and 75 by Prime Factorization
Prime factorization of 15, 25, 40, and 75 is (3 × 5) = 3^{1} × 5^{1}, (5 × 5) = 5^{2}, (2 × 2 × 2 × 5) = 2^{3} × 5^{1}, and (3 × 5 × 5) = 3^{1} × 5^{2} respectively. LCM of 15, 25, 40, and 75 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2^{3} × 3^{1} × 5^{2} = 600.
Hence, the LCM of 15, 25, 40, and 75 by prime factorization is 600.
LCM of 15, 25, 40, and 75 by Listing Multiples
To calculate the LCM of 15, 25, 40, 75 by listing out the common multiples, we can follow the given below steps:
 Step 1: List a few multiples of 15 (15, 30, 45, 60, 75 . . .), 25 (25, 50, 75, 100, 125 . . .), 40 (40, 80, 120, 160, 200 . . .), and 75 (75, 150, 225, 300, 375 . . .).
 Step 2: The common multiples from the multiples of 15, 25, 40, and 75 are 600, 1200, . . .
 Step 3: The smallest common multiple of 15, 25, 40, and 75 is 600.
∴ The least common multiple of 15, 25, 40, and 75 = 600.
ā Also Check:
 LCM of 3, 6 and 12  12
 LCM of 7 and 56  56
 LCM of 12 and 22  132
 LCM of 56 and 70  280
 LCM of 4 and 16  16
 LCM of 7 and 10  70
 LCM of 3 and 11  33
LCM of 15, 25, 40, and 75 Examples

Example 1: Find the smallest number which when divided by 15, 25, 40, and 75 leaves 4 as the remainder in each case.
Solution:
The smallest number exactly divisible by 15, 25, 40, and 75 = LCM(15, 25, 40, 75) ⇒ Smallest number which leaves 4 as remainder when divided by 15, 25, 40, and 75 = LCM(15, 25, 40, 75) + 4
 15 = 3^{1} × 5^{1}
 25 = 5^{2}
 40 = 2^{3} × 5^{1}
 75 = 3^{1} × 5^{2}
LCM(15, 25, 40, 75) = 2^{3} × 3^{1} × 5^{2} = 600
⇒ The required number = 600 + 4 = 604. 
Example 2: Find the smallest number that is divisible by 15, 25, 40, 75 exactly.
Solution:
The value of LCM(15, 25, 40, 75) will be the smallest number that is exactly divisible by 15, 25, 40, and 75.
⇒ Multiples of 15, 25, 40, and 75: Multiples of 15 = 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, . . . ., 540, 555, 570, 585, 600, . . . .
 Multiples of 25 = 25, 50, 75, 100, 125, 150, 175, 200, 225, 250, . . . ., 525, 550, 575, 600, . . . .
 Multiples of 40 = 40, 80, 120, 160, 200, 240, 280, 320, 360, 400, . . . ., 480, 520, 560, 600, . . . .
 Multiples of 75 = 75, 150, 225, 300, 375, 450, 525, 600, 675, 750, . . . ., 300, 375, 450, 525, 600, . . . .
Therefore, the LCM of 15, 25, 40, and 75 is 600.

Example 3: Which of the following is the LCM of 15, 25, 40, 75? 600, 24, 16, 40.
Solution:
The value of LCM of 15, 25, 40, and 75 is the smallest common multiple of 15, 25, 40, and 75. The number satisfying the given condition is 600. ∴LCM(15, 25, 40, 75) = 600.
FAQs on LCM of 15, 25, 40, and 75
What is the LCM of 15, 25, 40, and 75?
The LCM of 15, 25, 40, and 75 is 600. To find the least common multiple of 15, 25, 40, and 75, we need to find the multiples of 15, 25, 40, and 75 (multiples of 15 = 15, 30, 45, 60 . . . . 600 . . . . ; multiples of 25 = 25, 50, 75, 100 . . . . 600 . . . . ; multiples of 40 = 40, 80, 120, 160 . . . . 600 . . . . ; multiples of 75 = 75, 150, 225, 300 . . . . 600 . . . . ) and choose the smallest multiple that is exactly divisible by 15, 25, 40, and 75, i.e., 600.
What are the Methods to Find LCM of 15, 25, 40, 75?
The commonly used methods to find the LCM of 15, 25, 40, 75 are:
 Division Method
 Listing Multiples
 Prime Factorization Method
What is the Least Perfect Square Divisible by 15, 25, 40, and 75?
The least number divisible by 15, 25, 40, and 75 = LCM(15, 25, 40, 75)
LCM of 15, 25, 40, and 75 = 2 × 2 × 2 × 3 × 5 × 5 [Incomplete pair(s): 2, 3]
⇒ Least perfect square divisible by each 15, 25, 40, and 75 = LCM(15, 25, 40, 75) × 2 × 3 = 3600 [Square root of 3600 = √3600 = ±60]
Therefore, 3600 is the required number.
How to Find the LCM of 15, 25, 40, and 75 by Prime Factorization?
To find the LCM of 15, 25, 40, and 75 using prime factorization, we will find the prime factors, (15 = 3^{1} × 5^{1}), (25 = 5^{2}), (40 = 2^{3} × 5^{1}), and (75 = 3^{1} × 5^{2}). LCM of 15, 25, 40, and 75 is the product of prime factors raised to their respective highest exponent among the numbers 15, 25, 40, and 75.
⇒ LCM of 15, 25, 40, 75 = 2^{3} × 3^{1} × 5^{2} = 600.