A day full of math games & activities. Find one near you.

LCM of 12, 14, and 16

LCM of 12, 14, and 16 is the smallest number among all common multiples of 12, 14, and 16. The first few multiples of 12, 14, and 16 are (12, 24, 36, 48, 60 . . .), (14, 28, 42, 56, 70 . . .), and (16, 32, 48, 64, 80 . . .) respectively. There are 3 commonly used methods to find LCM of 12, 14, 16 - by listing multiples, by prime factorization, and by division method.

1.	LCM of 12, 14, and 16
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 12, 14, and 16?

Answer: LCM of 12, 14, and 16 is 336.

Explanation:

The LCM of three non-zero integers, a(12), b(14), and c(16), is the smallest positive integer m(336) that is divisible by a(12), b(14), and c(16) without any remainder.

Methods to Find LCM of 12, 14, and 16

The methods to find the LCM of 12, 14, and 16 are explained below.

By Division Method
By Listing Multiples
By Prime Factorization Method

LCM of 12, 14, and 16 by Division Method

To calculate the LCM of 12, 14, and 16 by the division method, we will divide the numbers(12, 14, 16) by their prime factors (preferably common). The product of these divisors gives the LCM of 12, 14, and 16.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 12, 14, and 16. Write this prime number(2) on the left of the given numbers(12, 14, and 16), separated as per the ladder arrangement.
Step 2: If any of the given numbers (12, 14, 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, 14, and 16 is the product of all prime numbers on the left, i.e. LCM(12, 14, 16) by division method = 2 × 2 × 2 × 2 × 3 × 7 = 336.

LCM of 12, 14, and 16 by Listing Multiples

To calculate the LCM of 12, 14, 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 . . .), 14 (14, 28, 42, 56, 70 . . .), and 16 (16, 32, 48, 64, 80 . . .).
Step 2: The common multiples from the multiples of 12, 14, and 16 are 336, 672, . . .
Step 3: The smallest common multiple of 12, 14, and 16 is 336.

∴ The least common multiple of 12, 14, and 16 = 336.

LCM of 12, 14, and 16 by Prime Factorization

Prime factorization of 12, 14, and 16 is (2 × 2 × 3) = 2² × 3¹, (2 × 7) = 2¹ × 7¹, and (2 × 2 × 2 × 2) = 2⁴ respectively. LCM of 12, 14, and 16 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2⁴ × 3¹ × 7¹ = 336.
Hence, the LCM of 12, 14, and 16 by prime factorization is 336.

☛ Also Check:

LCM of 12, 14, and 16 Examples

Example 1: Find the smallest number that is divisible by 12, 14, 16 exactly.

Solution:

The value of LCM(12, 14, 16) will be the smallest number that is exactly divisible by 12, 14, and 16.
⇒ Multiples of 12, 14, and 16:
- Multiples of 12 = 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, . . . ., 300, 312, 324, 336, . . . .
- Multiples of 14 = 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, . . . ., 280, 294, 308, 322, 336, . . . .
- Multiples of 16 = 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, . . . ., 304, 320, 336, . . . .
Therefore, the LCM of 12, 14, and 16 is 336.
Example 2: Verify the relationship between the GCD and LCM of 12, 14, and 16.

Solution:

The relation between GCD and LCM of 12, 14, and 16 is given as,
LCM(12, 14, 16) = [(12 × 14 × 16) × GCD(12, 14, 16)]/[GCD(12, 14) × GCD(14, 16) × GCD(12, 16)]
⇒ Prime factorization of 12, 14 and 16:
- 12 = 2² × 3¹
- 14 = 2¹ × 7¹
- 16 = 2⁴
∴ GCD of (12, 14), (14, 16), (12, 16) and (12, 14, 16) = 2, 2, 4 and 2 respectively.
Now, LHS = LCM(12, 14, 16) = 336.
And, RHS = [(12 × 14 × 16) × GCD(12, 14, 16)]/[GCD(12, 14) × GCD(14, 16) × GCD(12, 16)] = [(2688) × 2]/[2 × 2 × 4] = 336
LHS = RHS = 336.
Hence verified.
Example 3: Calculate the LCM of 12, 14, and 16 using the GCD of the given numbers.

Solution:

Prime factorization of 12, 14, 16:
- 12 = 2² × 3¹
- 14 = 2¹ × 7¹
- 16 = 2⁴
Therefore, GCD(12, 14) = 2, GCD(14, 16) = 2, GCD(12, 16) = 4, GCD(12, 14, 16) = 2
We know,
LCM(12, 14, 16) = [(12 × 14 × 16) × GCD(12, 14, 16)]/[GCD(12, 14) × GCD(14, 16) × GCD(12, 16)]
LCM(12, 14, 16) = (2688 × 2)/(2 × 2 × 4) = 336
⇒LCM(12, 14, 16) = 336

go to slide go to slide go to slide

Ready to see the world through math’s eyes?

Math is at the core of everything we do. Enjoy solving real-world math problems in live classes and become an expert at everything.

Book a Free Trial Class

FAQs on LCM of 12, 14, and 16

What is the LCM of 12, 14, and 16?

The LCM of 12, 14, and 16 is 336. To find the LCM of 12, 14, and 16, we need to find the multiples of 12, 14, and 16 (multiples of 12 = 12, 24, 36, 48 . . . . 336 . . . . ; multiples of 14 = 14, 28, 42, 56 . . . . 336 . . . . ; multiples of 16 = 16, 32, 48, 64 . . . . 336 . . . . ) and choose the smallest multiple that is exactly divisible by 12, 14, and 16, i.e., 336.

Which of the following is the LCM of 12, 14, and 16? 15, 10, 336, 36

The value of LCM of 12, 14, 16 is the smallest common multiple of 12, 14, and 16. The number satisfying the given condition is 336.

What is the Relation Between GCF and LCM of 12, 14, 16?

The following equation can be used to express the relation between GCF and LCM of 12, 14, 16, i.e. LCM(12, 14, 16) = [(12 × 14 × 16) × GCF(12, 14, 16)]/[GCF(12, 14) × GCF(14, 16) × GCF(12, 16)].

What are the Methods to Find LCM of 12, 14, 16?

The commonly used methods to find the LCM of 12, 14, 16 are:

Division Method
Listing Multiples
Prime Factorization Method

Download FREE Study Materials

LCM and GCF

Math worksheets and
visual curriculum