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LCM of 36, 60, and 72

LCM of 36, 60, and 72 is the smallest number among all common multiples of 36, 60, and 72. The first few multiples of 36, 60, and 72 are (36, 72, 108, 144, 180 . . .), (60, 120, 180, 240, 300 . . .), and (72, 144, 216, 288, 360 . . .) respectively. There are 3 commonly used methods to find LCM of 36, 60, 72 - by prime factorization, by division method, and by listing multiples.

1.	LCM of 36, 60, and 72
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is the LCM of 36, 60, and 72?

Answer: LCM of 36, 60, and 72 is 360.

Explanation:

The LCM of three non-zero integers, a(36), b(60), and c(72), is the smallest positive integer m(360) that is divisible by a(36), b(60), and c(72) without any remainder.

Methods to Find LCM of 36, 60, and 72

The methods to find the LCM of 36, 60, and 72 are explained below.

By Listing Multiples
By Prime Factorization Method
By Division Method

LCM of 36, 60, and 72 by Listing Multiples

To calculate the LCM of 36, 60, 72 by listing out the common multiples, we can follow the given below steps:

Step 1: List a few multiples of 36 (36, 72, 108, 144, 180 . . .), 60 (60, 120, 180, 240, 300 . . .), and 72 (72, 144, 216, 288, 360 . . .).
Step 2: The common multiples from the multiples of 36, 60, and 72 are 360, 720, . . .
Step 3: The smallest common multiple of 36, 60, and 72 is 360.

∴ The least common multiple of 36, 60, and 72 = 360.

LCM of 36, 60, and 72 by Prime Factorization

Prime factorization of 36, 60, and 72 is (2 × 2 × 3 × 3) = 2² × 3², (2 × 2 × 3 × 5) = 2² × 3¹ × 5¹, and (2 × 2 × 2 × 3 × 3) = 2³ × 3² respectively. LCM of 36, 60, and 72 can be obtained by multiplying prime factors raised to their respective highest power, i.e. 2³ × 3² × 5¹ = 360.
Hence, the LCM of 36, 60, and 72 by prime factorization is 360.

LCM of 36, 60, and 72 by Division Method

To calculate the LCM of 36, 60, and 72 by the division method, we will divide the numbers(36, 60, 72) by their prime factors (preferably common). The product of these divisors gives the LCM of 36, 60, and 72.

Step 1: Find the smallest prime number that is a factor of at least one of the numbers, 36, 60, and 72. Write this prime number(2) on the left of the given numbers(36, 60, and 72), separated as per the ladder arrangement.
Step 2: If any of the given numbers (36, 60, 72) 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 36, 60, and 72 is the product of all prime numbers on the left, i.e. LCM(36, 60, 72) by division method = 2 × 2 × 2 × 3 × 3 × 5 = 360.

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LCM of 36, 60, and 72 Examples

Example 1: Find the smallest number that is divisible by 36, 60, 72 exactly.

Solution:

The smallest number that is divisible by 36, 60, and 72 exactly is their LCM.
⇒ Multiples of 36, 60, and 72:
- Multiples of 36 = 36, 72, 108, 144, 180, 216, 252, 288, 324, 360, . . . .
- Multiples of 60 = 60, 120, 180, 240, 300, 360, . . . .
- Multiples of 72 = 72, 144, 216, 288, 360, . . . .
Therefore, the LCM of 36, 60, and 72 is 360.
Example 2: Verify the relationship between the GCD and LCM of 36, 60, and 72.

Solution:

The relation between GCD and LCM of 36, 60, and 72 is given as,
LCM(36, 60, 72) = [(36 × 60 × 72) × GCD(36, 60, 72)]/[GCD(36, 60) × GCD(60, 72) × GCD(36, 72)]
⇒ Prime factorization of 36, 60 and 72:
- 36 = 2² × 3²
- 60 = 2² × 3¹ × 5¹
- 72 = 2³ × 3²
∴ GCD of (36, 60), (60, 72), (36, 72) and (36, 60, 72) = 12, 12, 36 and 12 respectively.
Now, LHS = LCM(36, 60, 72) = 360.
And, RHS = [(36 × 60 × 72) × GCD(36, 60, 72)]/[GCD(36, 60) × GCD(60, 72) × GCD(36, 72)] = [(155520) × 12]/[12 × 12 × 36] = 360
LHS = RHS = 360.
Hence verified.
Example 3: Calculate the LCM of 36, 60, and 72 using the GCD of the given numbers.

Solution:

Prime factorization of 36, 60, 72:
- 36 = 2² × 3²
- 60 = 2² × 3¹ × 5¹
- 72 = 2³ × 3²
Therefore, GCD(36, 60) = 12, GCD(60, 72) = 12, GCD(36, 72) = 36, GCD(36, 60, 72) = 12
We know,
LCM(36, 60, 72) = [(36 × 60 × 72) × GCD(36, 60, 72)]/[GCD(36, 60) × GCD(60, 72) × GCD(36, 72)]
LCM(36, 60, 72) = (155520 × 12)/(12 × 12 × 36) = 360
⇒LCM(36, 60, 72) = 360

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FAQs on LCM of 36, 60, and 72

What is the LCM of 36, 60, and 72?

The LCM of 36, 60, and 72 is 360. To find the least common multiple (LCM) of 36, 60, and 72, we need to find the multiples of 36, 60, and 72 (multiples of 36 = 36, 72, 108, 144 . . . . 360 . . . . ; multiples of 60 = 60, 120, 180, 240, 360 . . . .; multiples of 72 = 72, 144, 216, 288 . . . . 360 . . . . ) and choose the smallest multiple that is exactly divisible by 36, 60, and 72, i.e., 360.

How to Find the LCM of 36, 60, and 72 by Prime Factorization?

To find the LCM of 36, 60, and 72 using prime factorization, we will find the prime factors, (36 = 2² × 3²), (60 = 2² × 3¹ × 5¹), and (72 = 2³ × 3²). LCM of 36, 60, and 72 is the product of prime factors raised to their respective highest exponent among the numbers 36, 60, and 72.
⇒ LCM of 36, 60, 72 = 2³ × 3² × 5¹ = 360.

What is the Least Perfect Square Divisible by 36, 60, and 72?

The least number divisible by 36, 60, and 72 = LCM(36, 60, 72)
LCM of 36, 60, and 72 = 2 × 2 × 2 × 3 × 3 × 5 [Incomplete pair(s): 2, 5]
⇒ Least perfect square divisible by each 36, 60, and 72 = LCM(36, 60, 72) × 2 × 5 = 3600 [Square root of 3600 = √3600 = ±60]
Therefore, 3600 is the required number.

What are the Methods to Find LCM of 36, 60, 72?

The commonly used methods to find the LCM of 36, 60, 72 are:

Division Method
Prime Factorization Method
Listing Multiples

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