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LCM of 6, 7, and 9

LCM of 6, 7, and 9 is the smallest number among all common multiples of 6, 7, and 9. The first few multiples of 6, 7, and 9 are (6, 12, 18, 24, 30 . . .), (7, 14, 21, 28, 35 . . .), and (9, 18, 27, 36, 45 . . .) respectively. There are 3 commonly used methods to find LCM of 6, 7, 9 - by listing multiples, by division method, and by prime factorization.

1.	LCM of 6, 7, and 9
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 6, 7, and 9?

Answer: LCM of 6, 7, and 9 is 126.

Explanation:

The LCM of three non-zero integers, a(6), b(7), and c(9), is the smallest positive integer m(126) that is divisible by a(6), b(7), and c(9) without any remainder.

Methods to Find LCM of 6, 7, and 9

Let's look at the different methods for finding the LCM of 6, 7, and 9.

By Division Method
By Prime Factorization Method
By Listing Multiples

LCM of 6, 7, and 9 by Division Method

To calculate the LCM of 6, 7, and 9 by the division method, we will divide the numbers(6, 7, 9) by their prime factors (preferably common). The product of these divisors gives the LCM of 6, 7, and 9.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 6, 7, and 9. Write this prime number(2) on the left of the given numbers(6, 7, and 9), separated as per the ladder arrangement.
Step 2: If any of the given numbers (6, 7, 9) 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 6, 7, and 9 is the product of all prime numbers on the left, i.e. LCM(6, 7, 9) by division method = 2 × 3 × 3 × 7 = 126.

LCM of 6, 7, and 9 by Prime Factorization

Prime factorization of 6, 7, and 9 is (2 × 3) = 2¹ × 3¹, (7) = 7¹, and (3 × 3) = 3² respectively. LCM of 6, 7, and 9 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2¹ × 3² × 7¹ = 126.
Hence, the LCM of 6, 7, and 9 by prime factorization is 126.

LCM of 6, 7, and 9 by Listing Multiples

To calculate the LCM of 6, 7, 9 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 6 (6, 12, 18, 24, 30 . . .), 7 (7, 14, 21, 28, 35 . . .), and 9 (9, 18, 27, 36, 45 . . .).
Step 2: The common multiples from the multiples of 6, 7, and 9 are 126, 252, . . .
Step 3: The smallest common multiple of 6, 7, and 9 is 126.

∴ The least common multiple of 6, 7, and 9 = 126.

☛ Also Check:

LCM of 6, 7, and 9 Examples

Example 1: Calculate the LCM of 6, 7, and 9 using the GCD of the given numbers.

Solution:

Prime factorization of 6, 7, 9:
- 6 = 2¹ × 3¹
- 7 = 7¹
- 9 = 3²
Therefore, GCD(6, 7) = 1, GCD(7, 9) = 1, GCD(6, 9) = 3, GCD(6, 7, 9) = 1
We know,
LCM(6, 7, 9) = [(6 × 7 × 9) × GCD(6, 7, 9)]/[GCD(6, 7) × GCD(7, 9) × GCD(6, 9)]
LCM(6, 7, 9) = (378 × 1)/(1 × 1 × 3) = 126
⇒LCM(6, 7, 9) = 126
Example 2: Verify the relationship between the GCD and LCM of 6, 7, and 9.

Solution:

The relation between GCD and LCM of 6, 7, and 9 is given as,
LCM(6, 7, 9) = [(6 × 7 × 9) × GCD(6, 7, 9)]/[GCD(6, 7) × GCD(7, 9) × GCD(6, 9)]
⇒ Prime factorization of 6, 7 and 9:
- 6 = 2¹ × 3¹
- 7 = 7¹
- 9 = 3²
∴ GCD of (6, 7), (7, 9), (6, 9) and (6, 7, 9) = 1, 1, 3 and 1 respectively.
Now, LHS = LCM(6, 7, 9) = 126.
And, RHS = [(6 × 7 × 9) × GCD(6, 7, 9)]/[GCD(6, 7) × GCD(7, 9) × GCD(6, 9)] = [(378) × 1]/[1 × 1 × 3] = 126
LHS = RHS = 126.
Hence verified.
Example 3: Find the smallest number that is divisible by 6, 7, 9 exactly.

Solution:

The value of LCM(6, 7, 9) will be the smallest number that is exactly divisible by 6, 7, and 9.
⇒ Multiples of 6, 7, and 9:
- Multiples of 6 = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, . . . ., 102, 108, 114, 120, 126, . . . .
- Multiples of 7 = 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, . . . ., 98, 105, 112, 119, 126, . . . .
- Multiples of 9 = 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, . . . ., 90, 99, 108, 117, 126, . . . .
Therefore, the LCM of 6, 7, and 9 is 126.

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FAQs on LCM of 6, 7, and 9

What is the LCM of 6, 7, and 9?

The LCM of 6, 7, and 9 is 126. To find the LCM of 6, 7, and 9, we need to find the multiples of 6, 7, and 9 (multiples of 6 = 6, 12, 18, 24 . . . . 126 . . . . ; multiples of 7 = 7, 14, 21, 28 . . . . 126 . . . . ; multiples of 9 = 9, 18, 27, 36 . . . . 126 . . . . ) and choose the smallest multiple that is exactly divisible by 6, 7, and 9, i.e., 126.

How to Find the LCM of 6, 7, and 9 by Prime Factorization?

To find the LCM of 6, 7, and 9 using prime factorization, we will find the prime factors, (6 = 2¹ × 3¹), (7 = 7¹), and (9 = 3²). LCM of 6, 7, and 9 is the product of prime factors raised to their respective highest exponent among the numbers 6, 7, and 9.
⇒ LCM of 6, 7, 9 = 2¹ × 3² × 7¹ = 126.

What is the Least Perfect Square Divisible by 6, 7, and 9?

The least number divisible by 6, 7, and 9 = LCM(6, 7, 9)
LCM of 6, 7, and 9 = 2 × 3 × 3 × 7 [Incomplete pair(s): 2, 7]
⇒ Least perfect square divisible by each 6, 7, and 9 = LCM(6, 7, 9) × 2 × 7 = 1764 [Square root of 1764 = √1764 = ±42]
Therefore, 1764 is the required number.

Which of the following is the LCM of 6, 7, and 9? 50, 35, 21, 126

The value of LCM of 6, 7, 9 is the smallest common multiple of 6, 7, and 9. The number satisfying the given condition is 126.

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