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LCM of 12 and 16

LCM of 12 and 16 is the smallest number among all common multiples of 12 and 16. The first few multiples of 12 and 16 are (12, 24, 36, 48, 60, 72, . . . ) and (16, 32, 48, 64, 80, 96, 112, . . . ) respectively. There are 3 commonly used methods to find LCM of 12 and 16 - by prime factorization, by listing multiples, and by division method.

1.	LCM of 12 and 16
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 12 and 16?

Answer: LCM of 12 and 16 is 48.

Explanation:

The LCM of two non-zero integers, x(12) and y(16), is the smallest positive integer m(48) that is divisible by both x(12) and y(16) without any remainder.

Methods to Find LCM of 12 and 16

Let's look at the different methods for finding the LCM of 12 and 16.

By Listing Multiples
By Division Method
By Prime Factorization Method

LCM of 12 and 16 by Listing Multiples

To calculate the LCM of 12 and 16 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 12 (12, 24, 36, 48, 60, 72, . . . ) and 16 (16, 32, 48, 64, 80, 96, 112, . . . . )
Step 2: The common multiples from the multiples of 12 and 16 are 48, 96, . . .
Step 3: The smallest common multiple of 12 and 16 is 48.

∴ The least common multiple of 12 and 16 = 48.

LCM of 12 and 16 by Division Method

To calculate the LCM of 12 and 16 by the division method, we will divide the numbers(12, 16) by their prime factors (preferably common). The product of these divisors gives the LCM of 12 and 16.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 12 and 16. Write this prime number(2) on the left of the given numbers(12 and 16), separated as per the ladder arrangement.
Step 2: If any of the given numbers (12, 16) 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 12 and 16 is the product of all prime numbers on the left, i.e. LCM(12, 16) by division method = 2 × 2 × 2 × 2 × 3 = 48.

LCM of 12 and 16 by Prime Factorization

Prime factorization of 12 and 16 is (2 × 2 × 3) = 2² × 3¹ and (2 × 2 × 2 × 2) = 2⁴ respectively. LCM of 12 and 16 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2⁴ × 3¹ = 48.
Hence, the LCM of 12 and 16 by prime factorization is 48.

☛ Also Check:

LCM of 12 and 16 Examples

Example 1: Verify the relationship between GCF and LCM of 12 and 16.

Solution:

The relation between GCF and LCM of 12 and 16 is given as,
LCM(12, 16) × GCF(12, 16) = Product of 12, 16
Prime factorization of 12 and 16 is given as, 12 = (2 × 2 × 3) = 2² × 3¹ and 16 = (2 × 2 × 2 × 2) = 2⁴
LCM(12, 16) = 48
GCF(12, 16) = 4
LHS = LCM(12, 16) × GCF(12, 16) = 48 × 4 = 192
RHS = Product of 12, 16 = 12 × 16 = 192
⇒ LHS = RHS = 192
Hence, verified.
Example 2: The product of two numbers is 192. If their GCD is 4, what is their LCM?

Solution:

Given: GCD = 4
product of numbers = 192
∵ LCM × GCD = product of numbers
⇒ LCM = Product/GCD = 192/4
Therefore, the LCM is 48.
The probable combination for the given case is LCM(12, 16) = 48.
Example 3: Find the smallest number that is divisible by 12 and 16 exactly.

Solution:

The smallest number that is divisible by 12 and 16 exactly is their LCM.
⇒ Multiples of 12 and 16:
- Multiples of 12 = 12, 24, 36, 48, 60, . . . .
- Multiples of 16 = 16, 32, 48, 64, 80, . . . .
Therefore, the LCM of 12 and 16 is 48.

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FAQs on LCM of 12 and 16

What is the LCM of 12 and 16?

The LCM of 12 and 16 is 48. To find the LCM (least common multiple) of 12 and 16, we need to find the multiples of 12 and 16 (multiples of 12 = 12, 24, 36, 48; multiples of 16 = 16, 32, 48, 64) and choose the smallest multiple that is exactly divisible by 12 and 16, i.e., 48.

If the LCM of 16 and 12 is 48, Find its GCF.

LCM(16, 12) × GCF(16, 12) = 16 × 12
Since the LCM of 16 and 12 = 48
⇒ 48 × GCF(16, 12) = 192
Therefore, the GCF = 192/48 = 4.

Which of the following is the LCM of 12 and 16? 35, 42, 48, 32

The value of LCM of 12, 16 is the smallest common multiple of 12 and 16. The number satisfying the given condition is 48.

How to Find the LCM of 12 and 16 by Prime Factorization?

To find the LCM of 12 and 16 using prime factorization, we will find the prime factors, (12 = 2 × 2 × 3) and (16 = 2 × 2 × 2 × 2). LCM of 12 and 16 is the product of prime factors raised to their respective highest exponent among the numbers 12 and 16.
⇒ LCM of 12, 16 = 2⁴ × 3¹ = 48.

What is the Least Perfect Square Divisible by 12 and 16?

The least number divisible by 12 and 16 = LCM(12, 16)
LCM of 12 and 16 = 2 × 2 × 2 × 2 × 3 [Incomplete pair(s): 3]
⇒ Least perfect square divisible by each 12 and 16 = LCM(12, 16) × 3 = 144 [Square root of 144 = √144 = ±12]
Therefore, 144 is the required number.

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