lcm of 12 and 15

LCM of 12 and 15 is 60. LCM stands for Least Common Multiple and is the least common multiple of both 12 and 15. LCM of 12 and 15 is calculated by division method, prime factorization, and by listing multiples methods.

In this article, we will discuss the concept of LCM and specifically explore its calculation for the numbers 6 and 8.

LCM of 12 and 15 is 60

What is the LCM of 12 and 15?

Answer: LCM of 6 and 8 is 24.

LCM of two non-zero integers, 12 and 15 is the smallest positive integer, 60 that is divisible by both 12 and 15 without any remainder. LCM of 6 and 8 is found using various methods that are added below:

What is LCM?

The smallest positive integer that can be divided by each of the provided numbers without leaving leftovers is the LCM (Least Common Multiple) of two or more numbers.

LCM of 12 and 15 Calculator

How to Find LCM of 12 and 15?

The three methods to find the LCM of 12 and 15 are:

• LCM of 12 and 15 by Prime Factorization
• LCM of 12 and 15 by Listing Multiple
• LCM of 12 and 15 by Long Division Method

LCM of 12 and 15 by Prime Factorization

To find the LCM of 12 and 15 by the prime factorization methods is found as:

• Prime factor of 12 = 2 × 2 × 3 or 2² × 3¹
• Prime factor of 15 = 3¹ × 5¹

LCM of 12 and 15 is 2 × 2 × 3 × 5 = 2² × 3 × 5 = 60

Therefore, LCM of 12 and 15 is 60

LCM of 12 and 15 by Listing Multiples

Least Common Multiple of 12 and 15, is found by listing multiple of 12 and 15 and then finding the smallest common multiple by comparing.

• Multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120
• Multiples of 15: 15, 30, 45, 60, 75, 90, 105, 120, 135, 150

Here, 60 is the first common multiple of both 12 and 15.

Therefore, LCM of 12 and 15 by listing multiples is 60.

• LCM of 12 and 15 by Long Division Method

By using the division method, follow the steps added below,

Step 1: Find the smallest prime number (2) that is a factor of at least one of the numbers. Whereas, 12 and 15 should be divided according to the ladder arrangement.

Step 2: If any of the given numbers (12, 15) are multiples of 2, divide them by 2 and record the quotient below. Reducing any number that is not divisible by the prime number is necessary.

Step 3: Continue this process, until 1 is left in the last row.

Thus, the LCM of 12 and 15 by long division method is 2 × 2 × 3 × 5 = 60.

LCM and HCF of 12 and 15

The Least Common Multiple(LCM) of 12 and 15 is 60 and the Highest Common Factor(HCF) of 12 and 15 is 3.

LCM × HCF = Product of Two Numbers

Prime factors of 12 and 15

• 12 = 2 × 2 × 3
• 15 = 3 × 5

LCM (12, 15) = 2 × 2 × 3 × 5 = 60

HCF (12, 15) = 3

12 × 15 = LCM (12, 15) × HCF (12, 15)

180 = 60 × 3

180 = 180

Example on LCM of 12 and 15

Example 1: Two numbers have GCD and LCM of 3 and 60, respectively. Find the other number if one of the numbers is 12.

Solution:

Let Y be the other number

GCD × LCM = 12 × Y

Y = (GCD × LCM)/12

Y = (3 × 60)/12

Y = 15

Hence, the other number is 15.

Example 2: Find the smallest number that 12 and 15 may be divided exactly.

Solution:

LCM is the smallest number that is exactly divisible by 12 and 15.

• Multiples of 12 = 12, 24, 36, 48, 60
• Multiples of 15 = 15, 30, 45, 60, 75

Common multiples in 12 and 15 is 60.

Therefore, the LCM of 12 and 15 is 60.

Practice Problems on LCM of 12 and 15

P1: Calculate the LCM of 12 and 15.

P2: Determine the smallest positive integer that is divisible by both 12 and 15.

P3: If x is the LCM of 12 and 15, what is the value of x?

P4: Find the LCM of 12 and 15 using prime factorization.

FAQs on LCM of 12 and 15

What is the Least Common Multiple (LCM) of 12 and 15?

LCM of 12 and 15 is 60.

How do you find the LCM of 12 and 15 by Prime Factorization?

Finding LCM of 12 and 15 by Prime Factorization method:

• Prime factors of number 12 = 2 × 2 × 3
• Prime factors of number 15 = 3 × 5

LCM = 2 × 2 × 3 × 5 = 60.

What is relation between GCD and LCM of 12 and 15?

Relation between 12 and 15 is, GCD (12, 15) × LCM (12, 15) = 12 × 15.

What are Methods to find LCM of 12 and 15?

Methods used to find the LCM of 12 and 15 are:

• Prime Factorization Method
• Long Division Method
• Listing Common Factors