LCM of 16, 18, and 24 is the smallest number among all common multiples of 16, 18, and 24. The first few multiples of 16, 18, and 24 are (16, 32, 48, 64, 80 . . .), (18, 36, 54, 72, 90 . . .), and (24, 48, 72, 96, 120 . . .) respectively. There are 3 commonly used methods to find LCM of 16, 18, 24 - by division method, by listing multiples, and by prime factorization.
What is the LCM of 16, 18, and 24?
Answer: LCM of 16, 18, and 24 is 144.
Explanation:
The LCM of three non-zero integers, a(16), b(18), and c(24), is the smallest positive integer m(144) that is divisible by a(16), b(18), and c(24) without any remainder.
Methods to Find LCM of 16, 18, and 24
The methods to find the LCM of 16, 18, and 24 are explained below.
- By Division Method
- By Listing Multiples
- By Prime Factorization Method
LCM of 16, 18, and 24 by Division Method
To calculate the LCM of 16, 18, and 24 by the division method, we will divide the numbers(16, 18, 24) by their prime factors (preferably common). The product of these divisors gives the LCM of 16, 18, and 24.
- Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 16, 18, and 24. Write this prime number(2) on the left of the given numbers(16, 18, and 24), separated as per the ladder arrangement.
- Step 2: If any of the given numbers (16, 18, 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, 18, and 24 is the product of all prime numbers on the left, i.e. LCM(16, 18, 24) by division method = 2 × 2 × 2 × 2 × 3 × 3 = 144.
LCM of 16, 18, and 24 by Listing Multiples
To calculate the LCM of 16, 18, 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 . . .), 18 (18, 36, 54, 72, 90 . . .), and 24 (24, 48, 72, 96, 120 . . .).
- Step 2: The common multiples from the multiples of 16, 18, and 24 are 144, 288, . . .
- Step 3: The smallest common multiple of 16, 18, and 24 is 144.
∴ The least common multiple of 16, 18, and 24 = 144.
LCM of 16, 18, and 24 by Prime Factorization
Prime factorization of 16, 18, and 24 is (2 × 2 × 2 × 2) = 24, (2 × 3 × 3) = 21 × 32, and (2 × 2 × 2 × 3) = 23 × 31 respectively. LCM of 16, 18, and 24 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 24 × 32 = 144.
Hence, the LCM of 16, 18, and 24 by prime factorization is 144.
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LCM of 16, 18, and 24 Examples
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Example 1: Verify the relationship between the GCD and LCM of 16, 18, and 24.
Solution:
The relation between GCD and LCM of 16, 18, and 24 is given as, LCM(16, 18, 24) = [(16 × 18 × 24) × GCD(16, 18, 24)]/[GCD(16, 18) × GCD(18, 24) × GCD(16, 24)]
⇒ Prime factorization of 16, 18 and 24:
- 16 = 24
- 18 = 21 × 32
- 24 = 23 × 31
∴ GCD of (16, 18), (18, 24), (16, 24) and (16, 18, 24) = 2, 6, 8 and 2 respectively. Now, LHS = LCM(16, 18, 24) = 144. And, RHS = [(16 × 18 × 24) × GCD(16, 18, 24)]/[GCD(16, 18) × GCD(18, 24) × GCD(16, 24)] = [(6912) × 2]/[2 × 6 × 8] = 144 LHS = RHS = 144.
Hence verified.
Example 2: Find the smallest number that is divisible by 16, 18, 24 exactly.
Solution:
The smallest number that is divisible by 16, 18, and 24 exactly is their LCM.
⇒ Multiples of 16, 18, and 24:
- Multiples of 16 = 16, 32, 48, 64, 80, 96, 112, 128, 144, . . . .
- Multiples of 18 = 18, 36, 54, 72, 90, 108, 126, 144, . . . .
- Multiples of 24 = 24, 48, 72, 96, 120, 144, 168, . . . .
Therefore, the LCM of 16, 18, and 24 is 144.
Example 3: Calculate the LCM of 16, 18, and 24 using the GCD of the given numbers.
Solution:
Prime factorization of 16, 18, 24:
- 16 = 24
- 18 = 21 × 32
- 24 = 23 × 31
Therefore, GCD(16, 18) = 2, GCD(18, 24) = 6, GCD(16, 24) = 8, GCD(16, 18, 24) = 2 We know, LCM(16, 18, 24) = [(16 × 18 × 24) × GCD(16, 18, 24)]/[GCD(16, 18) × GCD(18, 24) × GCD(16, 24)] LCM(16, 18, 24) = (6912 × 2)/(2 × 6 × 8) = 144
⇒LCM(16, 18, 24) = 144
Show Solution >
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The LCM of 16, 18, and 24 is 144. To find the LCM of 16, 18, and 24, we need to find the multiples of 16, 18, and 24 (multiples of 16 = 16, 32, 48, 64 . . . . 144 . . . . ; multiples of 18 = 18, 36, 54, 72 . . . . 144 . . . . ; multiples of 24 = 24, 48, 72, 96, 144 . . . .) and choose the smallest multiple that is exactly divisible by 16, 18, and 24, i.e., 144.
What is the Relation Between GCF and LCM of 16, 18, 24?
The following equation can be used to express the relation between GCF and LCM of 16, 18, 24, i.e. LCM(16, 18, 24) = [(16 × 18 × 24) × GCF(16, 18, 24)]/[GCF(16, 18) × GCF(18, 24) × GCF(16, 24)].
What are the Methods to Find LCM of 16, 18, 24?
The commonly used methods to find the LCM of 16, 18, 24 are:
- Division Method
- Listing Multiples
- Prime Factorization Method
What is the Least Perfect Square Divisible by 16, 18, and 24?
The least number divisible by 16, 18, and 24 = LCM(16, 18, 24)
LCM of 16, 18, and 24 = 2 × 2 × 2 × 2 × 3 × 3 [No incomplete pair]
⇒ Least perfect square divisible by each 16, 18, and 24 = LCM(16, 18, 24) = 144 [Square root of 144 = √144 = ±12]
Therefore, 144 is the required number.
Find the least number which is a perfect square and which is also divisible by 12,15 and 24 .
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Answer
Hint: Here we will first find the LCM of these three numbers. Then we will multiply the number that are not in pair in the factorization of the LCM because for a number to be a perfect square each factor should be in pair. We will simplify it further to get the required answer.
Complete step-by-step answer:
We will first find the LCM of these three numbers using the method of prime factorization.Now, we will first express each number in terms of the product of its prime factors.To write the prime factors, we should always start with the smallest prime number i.e. 2 and check divisibility. If the number is divisible by the prime number, then we can write the number as a product of the prime number and another number, which will be the quotient when the given number is divided by that prime number. Then we take the quotient and we will repeat the same process. This process is repeated until we get the quotient as 1. Now, we will find the factors of the first number 84.We know that 16 is an even number, so it can be written as $2 \times 8$ i.e.$16 = 2 \times 8$Now, we will further break the number 8 into its factors.$ \Rightarrow 16 = 2 \times 2 \times 2 \times 2$Similarly, we will find the factors of the second number 20.We know that 20 is an even number, so it can be written as $2 \times 10$ i.e.$20 = 2 \times 10$Now, we will further break the number 10 into its factors.$ \Rightarrow 20 = 2 \times 2 \times 5$Again we will find the factors of the third number 24.We know that 24 is an even number, so it can be written as $2 \times 12$ i.e.$24 = 2 \times 12$Now, we will further break the number 12 into its factors as we can see that it is a multiple of 2 and 3.Therefore, we can write it as $ \Rightarrow 24 = 2 \times 2 \times 2 \times 3$Now, to find the LCM, we need to multiply all the prime factors that has occurred the maximum number of time in either of the numbers. So, LCM of 16, 20 and 24 is a product of four 2, one 3 and one 5. As we can see that 2 have occurred four times in 24, 3 is occurring a maximum number of 1 time and 5 are occurring maximum number of one time.Therefore, $LCM\left( {16,20,24} \right) = 2 \times 2 \times 2 \times 2 \times 3 \times 5$ On multiplying the number, we get$ \Rightarrow LCM\left( {16,20,24} \right) = 240$ We know that for a number to be a perfect square each factor should be in pair of two or in even numbers we can see that 3 and 5 are not in pair of two. So to get the number, we will multiply 3 and 5 to the LCM obtained.Now we will find the least number that is perfect square and it is divisible by all these given numbers. The required least number $ = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5$On multiplying the numbers, we get$ \Rightarrow $ The required least number $ = 3600$Hence, the correct option is option D.
Note: Sometimes we get confused between factors of a number and multiples of a number. A factor is defined as a number which divides the given number completely but the multiple is defined as a number that is completely divided by the given number. Let’s take the number 10 as an example. We know that the factors of number 10 are 1, 2, 5 and 10 but the multiples of number 10 are 10, 20, 30, 40, and so on.