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HCF of 180, 252 and 324

HCF of 180, 252 and 324 is the largest possible number that divides 180, 252 and 324 exactly without any remainder. The factors of 180, 252 and 324 are (1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180), (1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252) and (1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 324) respectively. There are 3 commonly used methods to find the HCF of 180, 252 and 324 - Euclidean algorithm, prime factorization, and long division.

1.	HCF of 180, 252 and 324
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is HCF of 180, 252 and 324?

Answer: HCF of 180, 252 and 324 is 36.

Explanation:

The HCF of three non-zero integers, x(180), y(252) and z(324), is the highest positive integer m(36) that divides x(180), y(252) and z(324) without any remainder.

Methods to Find HCF of 180, 252 and 324

Let's look at the different methods for finding the HCF of 180, 252 and 324.

Long Division Method
Using Euclid's Algorithm
Listing Common Factors

HCF of 180, 252 and 324 by Long Division

HCF of 180, 252 and 324 can be represented as HCF of (HCF of 180, 252) and 324. HCF(180, 252, 324) can be thus calculated by first finding HCF(180, 252) using long division and thereafter using this result with 324 to perform long division again.

Step 1: Divide 252 (larger number) by 180 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (180) by the remainder (72). Repeat this process until the remainder = 0.
⇒ HCF(180, 252) = 36.
Step 3: Now to find the HCF of 36 and 324, we will perform a long division on 324 and 36.
Step 4: For remainder = 0, divisor = 36 ⇒ HCF(36, 324) = 36

Thus, HCF(180, 252, 324) = HCF(HCF(180, 252), 324) = 36.

HCF of 180, 252 and 324 by Euclidean Algorithm

As per the Euclidean Algorithm, HCF(X, Y) = HCF(Y, X mod Y)
where X > Y and mod is the modulo operator.

HCF(180, 252, 324) = HCF(HCF(180, 252), 324)

HCF(252, 180) = HCF(180, 252 mod 180) = HCF(180, 72)
HCF(180, 72) = HCF(72, 180 mod 72) = HCF(72, 36)
HCF(72, 36) = HCF(36, 72 mod 36) = HCF(36, 0)
HCF(36, 0) = 36 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Steps for HCF(36, 324)

HCF(324, 36) = HCF(36, 324 mod 36) = HCF(36, 0)
HCF(36, 0) = 36 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of HCF of 180, 252 and 324 is 36.

HCF of 180, 252 and 324 by Listing Common Factors

Factors of 180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180
Factors of 252: 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252
Factors of 324: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 324

There are 9 common factors of 180, 252 and 324, that are 1, 2, 3, 4, 36, 6, 9, 12, and 18. Therefore, the highest common factor of 180, 252 and 324 is 36.

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HCF of 180, 252 and 324 Examples

Example 1: Find the highest number that divides 180, 252, and 324 completely.

Solution:

The highest number that divides 180, 252, and 324 exactly is their highest common factor.
- Factors of 180 = 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180
- Factors of 252 = 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252
- Factors of 324 = 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 324
The HCF of 180, 252, and 324 is 36.
∴ The highest number that divides 180, 252, and 324 is 36.
Example 2: Calculate the HCF of 180, 252, and 324 using LCM of the given numbers.

Solution:

Prime factorization of 180, 252 and 324 is given as,
- 180 = 2 × 2 × 3 × 3 × 5
- 252 = 2 × 2 × 3 × 3 × 7
- 324 = 2 × 2 × 3 × 3 × 3 × 3
LCM(180, 252) = 1260, LCM(252, 324) = 2268, LCM(324, 180) = 1620, LCM(180, 252, 324) = 11340
⇒ HCF(180, 252, 324) = [(180 × 252 × 324) × LCM(180, 252, 324)]/[LCM(180, 252) × LCM (252, 324) × LCM(324, 180)]
⇒ HCF(180, 252, 324) = (14696640 × 11340)/(1260 × 2268 × 1620)
⇒ HCF(180, 252, 324) = 36.
Therefore, the HCF of 180, 252 and 324 is 36.
Example 3: Verify the relation between the LCM and HCF of 180, 252 and 324.

Solution:

The relation between the LCM and HCF of 180, 252 and 324 is given as, HCF(180, 252, 324) = [(180 × 252 × 324) × LCM(180, 252, 324)]/[LCM(180, 252) × LCM (252, 324) × LCM(180, 324)]
⇒ Prime factorization of 180, 252 and 324:
- 180 = 2 × 2 × 3 × 3 × 5
- 252 = 2 × 2 × 3 × 3 × 7
- 324 = 2 × 2 × 3 × 3 × 3 × 3
∴ LCM of (180, 252), (252, 324), (180, 324), and (180, 252, 324) is 1260, 2268, 1620, and 11340 respectively.
Now, LHS = HCF(180, 252, 324) = 36.
And, RHS = [(180 × 252 × 324) × LCM(180, 252, 324)]/[LCM(180, 252) × LCM (252, 324) × LCM(180, 324)] = [(14696640) × 11340]/[1260 × 2268 × 1620]
LHS = RHS = 36.
Hence verified.

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FAQs on HCF of 180, 252 and 324

What is the HCF of 180, 252 and 324?

The HCF of 180, 252 and 324 is 36. To calculate the highest common factor (HCF) of 180, 252 and 324, we need to factor each number (factors of 180 = 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180; factors of 252 = 1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252; factors of 324 = 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 324) and choose the highest factor that exactly divides 180, 252 and 324, i.e., 36.

Which of the following is HCF of 180, 252 and 324? 36, 337, 354, 346, 360, 339, 346

HCF of 180, 252, 324 will be the number that divides 180, 252, and 324 without leaving any remainder. The only number that satisfies the given condition is 36.

How to Find the HCF of 180, 252 and 324 by Prime Factorization?

To find the HCF of 180, 252 and 324, we will find the prime factorization of given numbers, i.e. 180 = 2 × 2 × 3 × 3 × 5; 252 = 2 × 2 × 3 × 3 × 7; 324 = 2 × 2 × 3 × 3 × 3 × 3.
⇒ Since 2, 2, 3, 3 are common terms in the prime factorization of 180, 252 and 324. Hence, HCF(180, 252, 324) = 2 × 2 × 3 × 3 = 36
☛ What is a Prime Number?

What is the Relation Between LCM and HCF of 180, 252 and 324?

The following equation can be used to express the relation between Least Common Multiple and HCF of 180, 252 and 324, i.e. HCF(180, 252, 324) = [(180 × 252 × 324) × LCM(180, 252, 324)]/[LCM(180, 252) × LCM (252, 324) × LCM(180, 324)].
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What are the Methods to Find HCF of 180, 252 and 324?

There are three commonly used methods to find the HCF of 180, 252 and 324.

By Prime Factorization
By Listing Common Factors
By Long Division

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