LCM of 21, 28, 36, and 45
LCM of 21, 28, 36, and 45 is the smallest number among all common multiples of 21, 28, 36, and 45. The first few multiples of 21, 28, 36, and 45 are (21, 42, 63, 84, 105 . . .), (28, 56, 84, 112, 140 . . .), (36, 72, 108, 144, 180 . . .), and (45, 90, 135, 180, 225 . . .) respectively. There are 3 commonly used methods to find LCM of 21, 28, 36, 45  by prime factorization, by division method, and by listing multiples.
1.  LCM of 21, 28, 36, and 45 
2.  List of Methods 
3.  Solved Examples 
4.  FAQs 
What is the LCM of 21, 28, 36, and 45?
Answer: LCM of 21, 28, 36, and 45 is 1260.
Explanation:
The LCM of four nonzero integers, a(21), b(28), c(36), and d(45), is the smallest positive integer m(1260) that is divisible by a(21), b(28), c(36), and d(45) without any remainder.
Methods to Find LCM of 21, 28, 36, and 45
Let's look at the different methods for finding the LCM of 21, 28, 36, and 45.
 By Prime Factorization Method
 By Division Method
 By Listing Multiples
LCM of 21, 28, 36, and 45 by Prime Factorization
Prime factorization of 21, 28, 36, and 45 is (3 × 7) = 3^{1} × 7^{1}, (2 × 2 × 7) = 2^{2} × 7^{1}, (2 × 2 × 3 × 3) = 2^{2} × 3^{2}, and (3 × 3 × 5) = 3^{2} × 5^{1} respectively. LCM of 21, 28, 36, and 45 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2^{2} × 3^{2} × 5^{1} × 7^{1} = 1260.
Hence, the LCM of 21, 28, 36, and 45 by prime factorization is 1260.
LCM of 21, 28, 36, and 45 by Division Method
To calculate the LCM of 21, 28, 36, and 45 by the division method, we will divide the numbers(21, 28, 36, 45) by their prime factors (preferably common). The product of these divisors gives the LCM of 21, 28, 36, and 45.
 Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 21, 28, 36, and 45. Write this prime number(2) on the left of the given numbers(21, 28, 36, and 45), separated as per the ladder arrangement.
 Step 2: If any of the given numbers (21, 28, 36, 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 21, 28, 36, and 45 is the product of all prime numbers on the left, i.e. LCM(21, 28, 36, 45) by division method = 2 × 2 × 3 × 3 × 5 × 7 = 1260.
LCM of 21, 28, 36, and 45 by Listing Multiples
To calculate the LCM of 21, 28, 36, 45 by listing out the common multiples, we can follow the given below steps:
 Step 1: List a few multiples of 21 (21, 42, 63, 84, 105 . . .), 28 (28, 56, 84, 112, 140 . . .), 36 (36, 72, 108, 144, 180 . . .), and 45 (45, 90, 135, 180, 225 . . .).
 Step 2: The common multiples from the multiples of 21, 28, 36, and 45 are 1260, 2520, . . .
 Step 3: The smallest common multiple of 21, 28, 36, and 45 is 1260.
∴ The least common multiple of 21, 28, 36, and 45 = 1260.
ā Also Check:
 LCM of 30 and 50  150
 LCM of 6, 8 and 9  72
 LCM of 15 and 45  45
 LCM of 40 and 50  200
 LCM of 35, 12 and 70  420
 LCM of 12 and 25  300
 LCM of 21 and 27  189
LCM of 21, 28, 36, and 45 Examples

Example 1: Find the smallest number that is divisible by 21, 28, 36, 45 exactly.
Solution:
The value of LCM(21, 28, 36, 45) will be the smallest number that is exactly divisible by 21, 28, 36, and 45.
⇒ Multiples of 21, 28, 36, and 45: Multiples of 21 = 21, 42, 63, 84, 105, 126, 147, 168, 189, 210, . . . ., 1218, 1239, 1260, . . . .
 Multiples of 28 = 28, 56, 84, 112, 140, 168, 196, 224, 252, 280, . . . ., 1148, 1176, 1204, 1232, 1260, . . . .
 Multiples of 36 = 36, 72, 108, 144, 180, 216, 252, 288, 324, 360, . . . ., 1152, 1188, 1224, 1260, . . . .
 Multiples of 45 = 45, 90, 135, 180, 225, 270, 315, 360, 405, 450, . . . ., 1125, 1170, 1215, 1260, . . . .
Therefore, the LCM of 21, 28, 36, and 45 is 1260.

Example 2: Find the smallest number which when divided by 21, 28, 36, and 45 leaves 3 as the remainder in each case.
Solution:
The smallest number exactly divisible by 21, 28, 36, and 45 = LCM(21, 28, 36, 45) ⇒ Smallest number which leaves 3 as remainder when divided by 21, 28, 36, and 45 = LCM(21, 28, 36, 45) + 3
 21 = 3^{1} × 7^{1}
 28 = 2^{2} × 7^{1}
 36 = 2^{2} × 3^{2}
 45 = 3^{2} × 5^{1}
LCM(21, 28, 36, 45) = 2^{2} × 3^{2} × 5^{1} × 7^{1} = 1260
⇒ The required number = 1260 + 3 = 1263. 
Example 3: Which of the following is the LCM of 21, 28, 36, 45? 21, 96, 1260, 18.
Solution:
The value of LCM of 21, 28, 36, and 45 is the smallest common multiple of 21, 28, 36, and 45. The number satisfying the given condition is 1260. ∴LCM(21, 28, 36, 45) = 1260.
FAQs on LCM of 21, 28, 36, and 45
What is the LCM of 21, 28, 36, and 45?
The LCM of 21, 28, 36, and 45 is 1260. To find the least common multiple (LCM) of 21, 28, 36, and 45, we need to find the multiples of 21, 28, 36, and 45 (multiples of 21 = 21, 42, 63, 84 . . . . 1260 . . . . ; multiples of 28 = 28, 56, 84, 112 . . . . 1260 . . . . ; multiples of 36 = 36, 72, 108, 144 . . . . 1260 . . . . ; multiples of 45 = 45, 90, 135, 180 . . . . 1260 . . . . ) and choose the smallest multiple that is exactly divisible by 21, 28, 36, and 45, i.e., 1260.
Which of the following is the LCM of 21, 28, 36, and 45? 40, 42, 35, 1260
The value of LCM of 21, 28, 36, 45 is the smallest common multiple of 21, 28, 36, and 45. The number satisfying the given condition is 1260.
How to Find the LCM of 21, 28, 36, and 45 by Prime Factorization?
To find the LCM of 21, 28, 36, and 45 using prime factorization, we will find the prime factors, (21 = 3^{1} × 7^{1}), (28 = 2^{2} × 7^{1}), (36 = 2^{2} × 3^{2}), and (45 = 3^{2} × 5^{1}). LCM of 21, 28, 36, and 45 is the product of prime factors raised to their respective highest exponent among the numbers 21, 28, 36, and 45.
⇒ LCM of 21, 28, 36, 45 = 2^{2} × 3^{2} × 5^{1} × 7^{1} = 1260.
What are the Methods to Find LCM of 21, 28, 36, 45?
The commonly used methods to find the LCM of 21, 28, 36, 45 are:
 Prime Factorization Method
 Division Method
 Listing Multiples
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