LCM of 40, 48, and 45
LCM of 40, 48, and 45 is the smallest number among all common multiples of 40, 48, and 45. The first few multiples of 40, 48, and 45 are (40, 80, 120, 160, 200 . . .), (48, 96, 144, 192, 240 . . .), and (45, 90, 135, 180, 225 . . .) respectively. There are 3 commonly used methods to find LCM of 40, 48, 45  by listing multiples, by division method, and by prime factorization.
1.  LCM of 40, 48, and 45 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is the LCM of 40, 48, and 45?
Answer: LCM of 40, 48, and 45 is 720.
Explanation:
The LCM of three nonzero integers, a(40), b(48), and c(45), is the smallest positive integer m(720) that is divisible by a(40), b(48), and c(45) without any remainder.
Methods to Find LCM of 40, 48, and 45
The methods to find the LCM of 40, 48, and 45 are explained below.
 By Division Method
 By Listing Multiples
 By Prime Factorization Method
LCM of 40, 48, and 45 by Division Method
To calculate the LCM of 40, 48, and 45 by the division method, we will divide the numbers(40, 48, 45) by their prime factors (preferably common). The product of these divisors gives the LCM of 40, 48, and 45.
 Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 40, 48, and 45. Write this prime number(2) on the left of the given numbers(40, 48, and 45), separated as per the ladder arrangement.
 Step 2: If any of the given numbers (40, 48, 45) 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 40, 48, and 45 is the product of all prime numbers on the left, i.e. LCM(40, 48, 45) by division method = 2 × 2 × 2 × 2 × 3 × 3 × 5 = 720.
LCM of 40, 48, and 45 by Listing Multiples
To calculate the LCM of 40, 48, 45 by listing out the common multiples, we can follow the given below steps:
 Step 1: List a few multiples of 40 (40, 80, 120, 160, 200 . . .), 48 (48, 96, 144, 192, 240 . . .), and 45 (45, 90, 135, 180, 225 . . .).
 Step 2: The common multiples from the multiples of 40, 48, and 45 are 720, 1440, . . .
 Step 3: The smallest common multiple of 40, 48, and 45 is 720.
∴ The least common multiple of 40, 48, and 45 = 720.
LCM of 40, 48, and 45 by Prime Factorization
Prime factorization of 40, 48, and 45 is (2 × 2 × 2 × 5) = 2^{3} × 5^{1}, (2 × 2 × 2 × 2 × 3) = 2^{4} × 3^{1}, and (3 × 3 × 5) = 3^{2} × 5^{1} respectively. LCM of 40, 48, and 45 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2^{4} × 3^{2} × 5^{1} = 720.
Hence, the LCM of 40, 48, and 45 by prime factorization is 720.
ā Also Check:
 LCM of 3, 9 and 15  45
 LCM of 15 and 27  135
 LCM of 4 and 10  20
 LCM of 2, 4, 6 and 8  24
 LCM of 8 and 56  56
 LCM of 30 and 36  180
 LCM of 42 and 72  504
LCM of 40, 48, and 45 Examples

Example 1: Verify the relationship between the GCD and LCM of 40, 48, and 45.
Solution:
The relation between GCD and LCM of 40, 48, and 45 is given as,
LCM(40, 48, 45) = [(40 × 48 × 45) × GCD(40, 48, 45)]/[GCD(40, 48) × GCD(48, 45) × GCD(40, 45)]
⇒ Prime factorization of 40, 48 and 45: 40 = 2^{3} × 5^{1}
 48 = 2^{4} × 3^{1}
 45 = 3^{2} × 5^{1}
∴ GCD of (40, 48), (48, 45), (40, 45) and (40, 48, 45) = 8, 3, 5 and 1 respectively.
Now, LHS = LCM(40, 48, 45) = 720.
And, RHS = [(40 × 48 × 45) × GCD(40, 48, 45)]/[GCD(40, 48) × GCD(48, 45) × GCD(40, 45)] = [(86400) × 1]/[8 × 3 × 5] = 720
LHS = RHS = 720.
Hence verified. 
Example 2: Calculate the LCM of 40, 48, and 45 using the GCD of the given numbers.
Solution:
Prime factorization of 40, 48, 45:
 40 = 2^{3} × 5^{1}
 48 = 2^{4} × 3^{1}
 45 = 3^{2} × 5^{1}
Therefore, GCD(40, 48) = 8, GCD(48, 45) = 3, GCD(40, 45) = 5, GCD(40, 48, 45) = 1
We know,
LCM(40, 48, 45) = [(40 × 48 × 45) × GCD(40, 48, 45)]/[GCD(40, 48) × GCD(48, 45) × GCD(40, 45)]
LCM(40, 48, 45) = (86400 × 1)/(8 × 3 × 5) = 720
⇒LCM(40, 48, 45) = 720 
Example 3: Find the smallest number that is divisible by 40, 48, 45 exactly.
Solution:
The value of LCM(40, 48, 45) will be the smallest number that is exactly divisible by 40, 48, and 45.
⇒ Multiples of 40, 48, and 45: Multiples of 40 = 40, 80, 120, 160, 200, 240, 280, 320, 360, 400, . . . ., 600, 640, 680, 720, . . . .
 Multiples of 48 = 48, 96, 144, 192, 240, 288, 336, 384, 432, 480, . . . ., 528, 576, 624, 672, 720, . . . .
 Multiples of 45 = 45, 90, 135, 180, 225, 270, 315, 360, 405, 450, . . . ., 540, 585, 630, 675, 720, . . . .
Therefore, the LCM of 40, 48, and 45 is 720.
FAQs on LCM of 40, 48, and 45
What is the LCM of 40, 48, and 45?
The LCM of 40, 48, and 45 is 720. To find the LCM (least common multiple) of 40, 48, and 45, we need to find the multiples of 40, 48, and 45 (multiples of 40 = 40, 80, 120, 160 . . . . 720 . . . . ; multiples of 48 = 48, 96, 144, 192 . . . . 720 . . . . ; multiples of 45 = 45, 90, 135, 180 . . . . 720 . . . . ) and choose the smallest multiple that is exactly divisible by 40, 48, and 45, i.e., 720.
How to Find the LCM of 40, 48, and 45 by Prime Factorization?
To find the LCM of 40, 48, and 45 using prime factorization, we will find the prime factors, (40 = 2^{3} × 5^{1}), (48 = 2^{4} × 3^{1}), and (45 = 3^{2} × 5^{1}). LCM of 40, 48, and 45 is the product of prime factors raised to their respective highest exponent among the numbers 40, 48, and 45.
⇒ LCM of 40, 48, 45 = 2^{4} × 3^{2} × 5^{1} = 720.
What are the Methods to Find LCM of 40, 48, 45?
The commonly used methods to find the LCM of 40, 48, 45 are:
 Prime Factorization Method
 Listing Multiples
 Division Method
What is the Least Perfect Square Divisible by 40, 48, and 45?
The least number divisible by 40, 48, and 45 = LCM(40, 48, 45)
LCM of 40, 48, and 45 = 2 × 2 × 2 × 2 × 3 × 3 × 5 [Incomplete pair(s): 5]
⇒ Least perfect square divisible by each 40, 48, and 45 = LCM(40, 48, 45) × 5 = 3600 [Square root of 3600 = √3600 = ±60]
Therefore, 3600 is the required number.
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