What is the least common multiple of 9 and 12

What is the LCM and GCF of 9 and 12?

The question "What is the LCM and GCF of 9 and 12?" can be split into two questions: "What is the LCM of 9 and 12?" and "What is the GCF of 9 and 12?" In the question "What is the LCM and GCF of 9 and 12?", LCM is the abbreviation of Least Common Multiple and GCF is the abbreviation of Greatest Common Factor. To find the LCM, we first list the multiples of 9 and 12 and then we find the smallest multiple they have in common. To find the multiples of any number, you simply multiply the number by 1, then by 2, then by 3 and so on. Here is the beginning list of multiples of 9 and 12:

Multiples of 9: 9, 18, 27, 36, 45, 54, etc.

Multiples of 12: 12, 24, 36, 48, 60, 72, etc.

The least multiple on the two lists that they have in common is the LCM of 9 and 12. Therefore, the LCM of 9 and 12 is 36. To find the GCF, we first list the factors of 9 and 12 and then we find the largest factor they have in common. The factors of any number, are all the numbers that you can evenly divide into that number. In other words, the factors of 9 are all the numbers that can evenly divide into 9, and the factors of 12 are all the numbers that can evenly divide into 12. Here are the factors for 9 and 12:

Factors of 9: 1, 3, and 9.

Factors of 12: 1, 2, 3, 4, 6, and 12.

The greatest factor on the two lists that they have in common is the GCF of 9 and 12. Therefore, the GCF of 9 and 12 is 3.

In summary, the answer to the question "What is the LCM and GCF of 9 and 12?" is 36 and 3.

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On this page below, we'll use the Prime Factorisation, and the List of Multiples method to find out the LCM of 9 and 12.

Follow the steps below, and let's calculate the LCM of 9 and 12.

Method 1 - Prime factorization

Step 1: Create a list of all the prime factors of the numbers 9 and 12:

The prime factors of 9 are 3 and 3. Prime factorization of 9 in exponential form is:

9 = 32

The prime factors of 12 are 2, 2 and 3. Prime factorization of 12 in exponential form is:

12 = 22x31

Step 2: Identify the highest power of each prime number from the above boxes:

Step 3: Multiply these values together:

Step 4: The result:

As seen on the calculation above, we have now obtained the LCM of 9 and 12.

The Least Common Multiple of 9 and 12 is 36.

Method 2 - List of Multiples

Find and list multiples of each number until the first common multiple is found. This is the lowest common multiple.

Multiples of 9:

9, 18, 27, 36, 45, 54, 63

Multiples of 12:

Therefore,

The full form of LCM is Least Common Multiple and also noted as Lowest Common Multiple (LCM) or Least Common Divisor (LCD). LCM is the smallest positive integer that is equally divisible by two integers a and b. It is denoted as LCM(a,b). For example, if you take integer a=8 and b=12, the LCM(a,b) i.e., LCM(8,12)=24.

Use this free online LCM Calculator and provide the input numbers that are given to calculate the LCM of two integers. Give the first input in the number 1 field and then second input in the number 2 field and press on the ‘Calculate’ button which is colored in blue. Remember not to use commas within your numbers like 5,000, 7,500.

Ex: LCM of 44 and 60 (or) LCM of 45 and 30 (or) LCM of 32 and 48

Here are some samples of LCM of two numbers calculations.

The LCM of 9 and 12 is 36.

Steps to find LCM

1. Find the prime factorization of 9
   9 = 3 × 3
2. Find the prime factorization of 12
   12 = 2 × 2 × 3
3. Multiply each factor the greater number of times it occurs in steps i) or ii) above to find the LCM:

   LCM = 2 × 2 × 3 × 3

4. LCM = 36

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Find least common multiple (LCM) of: 18 & 24 27 & 36 3 & 4 45 & 60 63 & 84 18 & 12 9 & 24 27 & 12 9 & 36 45 & 12 9 & 60 63 & 12 9 & 84

Enter two numbers separate by comma. To find least common multiple (LCM) of more than two numbers, click here.

Least common multiple or lowest common denominator (lcd) can be calculated in two way; with the LCM formula calculation of greatest common factor (GCF), or multiplying the prime factors with the highest exponent factor.

Least common multiple (LCM) of 9 and 12 is 36.

LCM(9,12) = 36

Least Common Multiple of 9 and 12 with GCF Formula

The formula of LCM is LCM(a,b) = ( a × b) / GCF(a,b).
We need to calculate greatest common factor 9 and 12, than apply into the LCM equation.

GCF(9,12) = 3 LCM(9,12) = ( 9 × 12) / 3 LCM(9,12) = 108 / 3

LCM(9,12) = 36

Least Common Multiple (LCM) of 9 and 12 with Primes

Least common multiple can be found by multiplying the highest exponent prime factors of 9 and 12. First we will calculate the prime factors of 9 and 12.

Prime Factorization of 9

Prime factors of 9 are 3. Prime factorization of 9 in exponential form is:

9 = 32

Prime Factorization of 12

Prime factors of 12 are 2, 3. Prime factorization of 12 in exponential form is:

12 = 22 × 31

Now multiplying the highest exponent prime factors to calculate the LCM of 9 and 12.

LCM(9,12) = 32 × 22
LCM(9,12) = 36

LCM of 9 and 12 is the smallest number among all common multiples of 9 and 12. The first few multiples of 9 and 12 are (9, 18, 27, 36, 45, . . . ) and (12, 24, 36, 48, 60, . . . ) respectively. There are 3 commonly used methods to find LCM of 9 and 12 - by division method, by listing multiples, and by prime factorization.

What is the LCM of 9 and 12?

Answer: LCM of 9 and 12 is 36.

Explanation:

The LCM of two non-zero integers, x(9) and y(12), is the smallest positive integer m(36) that is divisible by both x(9) and y(12) without any remainder.

Methods to Find LCM of 9 and 12

The methods to find the LCM of 9 and 12 are explained below.

• By Listing Multiples
• By Division Method
• By Prime Factorization Method

LCM of 9 and 12 by Listing Multiples

To calculate the LCM of 9 and 12 by listing out the common multiples, we can follow the given below steps:

• Step 1: List a few multiples of 9 (9, 18, 27, 36, 45, . . . ) and 12 (12, 24, 36, 48, 60, . . . . )
• Step 2: The common multiples from the multiples of 9 and 12 are 36, 72, . . .
• Step 3: The smallest common multiple of 9 and 12 is 36.

∴ The least common multiple of 9 and 12 = 36.

LCM of 9 and 12 by Division Method

To calculate the LCM of 9 and 12 by the division method, we will divide the numbers(9, 12) by their prime factors (preferably common). The product of these divisors gives the LCM of 9 and 12.

• Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 9 and 12. Write this prime number(2) on the left of the given numbers(9 and 12), separated as per the ladder arrangement.
• Step 2: If any of the given numbers (9, 12) 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 9 and 12 is the product of all prime numbers on the left, i.e. LCM(9, 12) by division method = 2 × 2 × 3 × 3 = 36.

LCM of 9 and 12 by Prime Factorization

Prime factorization of 9 and 12 is (3 × 3) = 32 and (2 × 2 × 3) = 22 × 31 respectively. LCM of 9 and 12 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 22 × 32 = 36.
Hence, the LCM of 9 and 12 by prime factorization is 36.

☛ Also Check:

1. Example 1: Verify the relationship between GCF and LCM of 9 and 12.

   Solution:

   The relation between GCF and LCM of 9 and 12 is given as, LCM(9, 12) × GCF(9, 12) = Product of 9, 12

   Prime factorization of 9 and 12 is given as, 9 = (3 × 3) = 32 and 12 = (2 × 2 × 3) = 22 × 31

   LCM(9, 12) = 36 GCF(9, 12) = 3 LHS = LCM(9, 12) × GCF(9, 12) = 36 × 3 = 108 RHS = Product of 9, 12 = 9 × 12 = 108 ⇒ LHS = RHS = 108

   Hence, verified.

Example 2: The product of two numbers is 108. If their GCD is 3, what is their LCM?

Solution:

Given: GCD = 3 product of numbers = 108 ∵ LCM × GCD = product of numbers ⇒ LCM = Product/GCD = 108/3 Therefore, the LCM is 36.

The probable combination for the given case is LCM(9, 12) = 36.

Example 3: The GCD and LCM of two numbers are 3 and 36 respectively. If one number is 12, find the other number.

Solution:

Let the other number be y.
∵ GCD × LCM = 12 × y ⇒ y = (GCD × LCM)/12 ⇒ y = (3 × 36)/12 ⇒ y = 9

Therefore, the other number is 9.

Show Solution >

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The LCM of 9 and 12 is 36. To find the least common multiple of 9 and 12, we need to find the multiples of 9 and 12 (multiples of 9 = 9, 18, 27, 36; multiples of 12 = 12, 24, 36, 48) and choose the smallest multiple that is exactly divisible by 9 and 12, i.e., 36.

What is the Relation Between GCF and LCM of 9, 12?

The following equation can be used to express the relation between GCF and LCM of 9 and 12, i.e. GCF × LCM = 9 × 12.

Which of the following is the LCM of 9 and 12? 50, 15, 45, 36

The value of LCM of 9, 12 is the smallest common multiple of 9 and 12. The number satisfying the given condition is 36.

If the LCM of 12 and 9 is 36, Find its GCF.

LCM(12, 9) × GCF(12, 9) = 12 × 9 Since the LCM of 12 and 9 = 36 ⇒ 36 × GCF(12, 9) = 108

Therefore, the GCF = 108/36 = 3.

What are the Methods to Find LCM of 9 and 12?

The commonly used methods to find the LCM of 9 and 12 are:

• Prime Factorization Method
• Division Method
• Listing Multiples