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LCM of 72, 126, and 168

LCM of 72, 126, and 168 is the smallest number among all common multiples of 72, 126, and 168. The first few multiples of 72, 126, and 168 are (72, 144, 216, 288, 360 . . .), (126, 252, 378, 504, 630 . . .), and (168, 336, 504, 672, 840 . . .) respectively. There are 3 commonly used methods to find LCM of 72, 126, 168 - by division method, by listing multiples, and by prime factorization.

1.	LCM of 72, 126, and 168
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 72, 126, and 168?

Answer: LCM of 72, 126, and 168 is 504.

Explanation:

The LCM of three non-zero integers, a(72), b(126), and c(168), is the smallest positive integer m(504) that is divisible by a(72), b(126), and c(168) without any remainder.

Methods to Find LCM of 72, 126, and 168

Let's look at the different methods for finding the LCM of 72, 126, and 168.

By Prime Factorization Method
By Division Method
By Listing Multiples

LCM of 72, 126, and 168 by Prime Factorization

Prime factorization of 72, 126, and 168 is (2 × 2 × 2 × 3 × 3) = 2³ × 3², (2 × 3 × 3 × 7) = 2¹ × 3² × 7¹, and (2 × 2 × 2 × 3 × 7) = 2³ × 3¹ × 7¹ respectively. LCM of 72, 126, and 168 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2³ × 3² × 7¹ = 504.
Hence, the LCM of 72, 126, and 168 by prime factorization is 504.

LCM of 72, 126, and 168 by Division Method

To calculate the LCM of 72, 126, and 168 by the division method, we will divide the numbers(72, 126, 168) by their prime factors (preferably common). The product of these divisors gives the LCM of 72, 126, and 168.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 72, 126, and 168. Write this prime number(2) on the left of the given numbers(72, 126, and 168), separated as per the ladder arrangement.
Step 2: If any of the given numbers (72, 126, 168) 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 72, 126, and 168 is the product of all prime numbers on the left, i.e. LCM(72, 126, 168) by division method = 2 × 2 × 2 × 3 × 3 × 7 = 504.

LCM of 72, 126, and 168 by Listing Multiples

To calculate the LCM of 72, 126, 168 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 72 (72, 144, 216, 288, 360 . . .), 126 (126, 252, 378, 504, 630 . . .), and 168 (168, 336, 504, 672, 840 . . .).
Step 2: The common multiples from the multiples of 72, 126, and 168 are 504, 1008, . . .
Step 3: The smallest common multiple of 72, 126, and 168 is 504.

∴ The least common multiple of 72, 126, and 168 = 504.

☛ Also Check:

LCM of 72, 126, and 168 Examples

Example 1: Calculate the LCM of 72, 126, and 168 using the GCD of the given numbers.

Solution:

Prime factorization of 72, 126, 168:
- 72 = 2³ × 3²
- 126 = 2¹ × 3² × 7¹
- 168 = 2³ × 3¹ × 7¹
Therefore, GCD(72, 126) = 18, GCD(126, 168) = 42, GCD(72, 168) = 24, GCD(72, 126, 168) = 6
We know,
LCM(72, 126, 168) = [(72 × 126 × 168) × GCD(72, 126, 168)]/[GCD(72, 126) × GCD(126, 168) × GCD(72, 168)]
LCM(72, 126, 168) = (1524096 × 6)/(18 × 42 × 24) = 504
⇒LCM(72, 126, 168) = 504
Example 2: Find the smallest number that is divisible by 72, 126, 168 exactly.

Solution:

The smallest number that is divisible by 72, 126, and 168 exactly is their LCM.
⇒ Multiples of 72, 126, and 168:
- Multiples of 72 = 72, 144, 216, 288, 360, 432, 504, . . . .
- Multiples of 126 = 126, 252, 378, 504, 630, 756, . . . .
- Multiples of 168 = 168, 336, 504, 672, 840, 1008, . . . .
Therefore, the LCM of 72, 126, and 168 is 504.
Example 3: Verify the relationship between the GCD and LCM of 72, 126, and 168.

Solution:

The relation between GCD and LCM of 72, 126, and 168 is given as,
LCM(72, 126, 168) = [(72 × 126 × 168) × GCD(72, 126, 168)]/[GCD(72, 126) × GCD(126, 168) × GCD(72, 168)]
⇒ Prime factorization of 72, 126 and 168:
- 72 = 2³ × 3²
- 126 = 2¹ × 3² × 7¹
- 168 = 2³ × 3¹ × 7¹
∴ GCD of (72, 126), (126, 168), (72, 168) and (72, 126, 168) = 18, 42, 24 and 6 respectively.
Now, LHS = LCM(72, 126, 168) = 504.
And, RHS = [(72 × 126 × 168) × GCD(72, 126, 168)]/[GCD(72, 126) × GCD(126, 168) × GCD(72, 168)] = [(1524096) × 6]/[18 × 42 × 24] = 504
LHS = RHS = 504.
Hence verified.

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FAQs on LCM of 72, 126, and 168

What is the LCM of 72, 126, and 168?

The LCM of 72, 126, and 168 is 504. To find the LCM of 72, 126, and 168, we need to find the multiples of 72, 126, and 168 (multiples of 72 = 72, 144, 216, 288, 432, 504 . . . .; multiples of 126 = 126, 252, 378, 504 . . . .; multiples of 168 = 168, 336, 504, 672 . . . .) and choose the smallest multiple that is exactly divisible by 72, 126, and 168, i.e., 504.

Which of the following is the LCM of 72, 126, and 168? 25, 504, 12, 110

The value of LCM of 72, 126, 168 is the smallest common multiple of 72, 126, and 168. The number satisfying the given condition is 504.

What are the Methods to Find LCM of 72, 126, 168?

The commonly used methods to find the LCM of 72, 126, 168 are:

Division Method
Prime Factorization Method
Listing Multiples

How to Find the LCM of 72, 126, and 168 by Prime Factorization?

To find the LCM of 72, 126, and 168 using prime factorization, we will find the prime factors, (72 = 2³ × 3²), (126 = 2¹ × 3² × 7¹), and (168 = 2³ × 3¹ × 7¹). LCM of 72, 126, and 168 is the product of prime factors raised to their respective highest exponent among the numbers 72, 126, and 168.
⇒ LCM of 72, 126, 168 = 2³ × 3² × 7¹ = 504.

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