LCM of 16, 20, and 24 is the smallest number among all common multiples of 16, 20, and 24. The first few multiples of 16, 20, and 24 are (16, 32, 48, 64, 80 . . .), (20, 40, 60, 80, 100 . . .), and (24, 48, 72, 96, 120 . . .) respectively. There are 3 commonly used methods to find LCM of 16, 20, 24 - by listing multiples, by division method, and by prime factorization.
What is the LCM of 16, 20, and 24?
Answer: LCM of 16, 20, and 24 is 240.
Explanation:
The LCM of three non-zero integers, a(16), b(20), and c(24), is the smallest positive integer m(240) that is divisible by a(16), b(20), and c(24) without any remainder.
Methods to Find LCM of 16, 20, and 24
The methods to find the LCM of 16, 20, and 24 are explained below.
- By Division Method
- By Prime Factorization Method
- By Listing Multiples
LCM of 16, 20, and 24 by Division Method
To calculate the LCM of 16, 20, and 24 by the division method, we will divide the numbers(16, 20, 24) by their prime factors (preferably common). The product of these divisors gives the LCM of 16, 20, and 24.
- Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 16, 20, and 24. Write this prime number(2) on the left of the given numbers(16, 20, and 24), separated as per the ladder arrangement.
- Step 2: If any of the given numbers (16, 20, 24) 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 16, 20, and 24 is the product of all prime numbers on the left, i.e. LCM(16, 20, 24) by division method = 2 × 2 × 2 × 2 × 3 × 5 = 240.
LCM of 16, 20, and 24 by Prime Factorization
Prime factorization of 16, 20, and 24 is (2 × 2 × 2 × 2) = 24, (2 × 2 × 5) = 22 × 51, and (2 × 2 × 2 × 3) = 23 × 31 respectively. LCM of 16, 20, and 24 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 24 × 31 × 51 = 240.
Hence, the LCM of 16, 20, and 24 by prime factorization is 240.
LCM of 16, 20, and 24 by Listing Multiples
To calculate the LCM of 16, 20, 24 by listing out the common multiples, we can follow the given below steps:
- Step 1: List a few multiples of 16 (16, 32, 48, 64, 80 . . .), 20 (20, 40, 60, 80, 100 . . .), and 24 (24, 48, 72, 96, 120 . . .).
- Step 2: The common multiples from the multiples of 16, 20, and 24 are 240, 480, . . .
- Step 3: The smallest common multiple of 16, 20, and 24 is 240.
∴ The least common multiple of 16, 20, and 24 = 240.
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LCM of 16, 20, and 24 Examples
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Example 1: Find the smallest number that is divisible by 16, 20, 24 exactly.
Solution:
The smallest number that is divisible by 16, 20, and 24 exactly is their LCM.
⇒ Multiples of 16, 20, and 24:- Multiples of 16 = 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240, . . . .
- Multiples of 20 = 20, 40, 60, 80, 100, 120, 140, 160, 180, 200, 220, 240, . . . .
- Multiples of 24 = 24, 48, 72, 96, 120, 144, 168, 192, 216, 240, . . . .
Therefore, the LCM of 16, 20, and 24 is 240.
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Example 2: Calculate the LCM of 16, 20, and 24 using the GCD of the given numbers.
Solution:
Prime factorization of 16, 20, 24:
- 16 = 24
- 20 = 22 × 51
- 24 = 23 × 31
Therefore, GCD(16, 20) = 4, GCD(20, 24) = 4, GCD(16, 24) = 8, GCD(16, 20, 24) = 4 We know, LCM(16, 20, 24) = [(16 × 20 × 24) × GCD(16, 20, 24)]/[GCD(16, 20) × GCD(20, 24) × GCD(16, 24)] LCM(16, 20, 24) = (7680 × 4)/(4 × 4 × 8) = 240
⇒LCM(16, 20, 24) = 240
Example 3: Verify the relationship between the GCD and LCM of 16, 20, and 24.
Solution:
The relation between GCD and LCM of 16, 20, and 24 is given as, LCM(16, 20, 24) = [(16 × 20 × 24) × GCD(16, 20, 24)]/[GCD(16, 20) × GCD(20, 24) × GCD(16, 24)]
⇒ Prime factorization of 16, 20 and 24:
- 16 = 24
- 20 = 22 × 51
- 24 = 23 × 31
∴ GCD of (16, 20), (20, 24), (16, 24) and (16, 20, 24) = 4, 4, 8 and 4 respectively. Now, LHS = LCM(16, 20, 24) = 240. And, RHS = [(16 × 20 × 24) × GCD(16, 20, 24)]/[GCD(16, 20) × GCD(20, 24) × GCD(16, 24)] = [(7680) × 4]/[4 × 4 × 8] = 240 LHS = RHS = 240.
Hence verified.
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The LCM of 16, 20, and 24 is 240. To find the LCM (least common multiple) of 16, 20, and 24, we need to find the multiples of 16, 20, and 24 (multiples of 16 = 16, 32, 48, 64 . . . . 240 . . . . ; multiples of 20 = 20, 40, 60, 80 . . . . 240 . . . . ; multiples of 24 = 24, 48, 72, 96 . . . . 240 . . . . ) and choose the smallest multiple that is exactly divisible by 16, 20, and 24, i.e., 240.
What is the Least Perfect Square Divisible by 16, 20, and 24?
The least number divisible by 16, 20, and 24 = LCM(16, 20, 24)
LCM of 16, 20, and 24 = 2 × 2 × 2 × 2 × 3 × 5 [Incomplete pair(s): 3, 5]
⇒ Least perfect square divisible by each 16, 20, and 24 = LCM(16, 20, 24) × 3 × 5 = 3600 [Square root of 3600 = √3600 = ±60]
Therefore, 3600 is the required number.
What are the Methods to Find LCM of 16, 20, 24?
The commonly used methods to find the LCM of 16, 20, 24 are:
- Prime Factorization Method
- Listing Multiples
- Division Method
What is the Relation Between GCF and LCM of 16, 20, 24?
The following equation can be used to express the relation between GCF and LCM of 16, 20, 24, i.e. LCM(16, 20, 24) = [(16 × 20 × 24) × GCF(16, 20, 24)]/[GCF(16, 20) × GCF(20, 24) × GCF(16, 24)].