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HCF of 144, 180 and 192

HCF of 144, 180 and 192 is the largest possible number that divides 144, 180 and 192 exactly without any remainder. The factors of 144, 180 and 192 are (1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144), (1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180) and (1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 192) respectively. There are 3 commonly used methods to find the HCF of 144, 180 and 192 - prime factorization, long division, and Euclidean algorithm.

1.	HCF of 144, 180 and 192
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is HCF of 144, 180 and 192?

Answer: HCF of 144, 180 and 192 is 12.

Explanation:

The HCF of three non-zero integers, x(144), y(180) and z(192), is the highest positive integer m(12) that divides x(144), y(180) and z(192) without any remainder.

Methods to Find HCF of 144, 180 and 192

The methods to find the HCF of 144, 180 and 192 are explained below.

Using Euclid's Algorithm
Listing Common Factors
Long Division Method

HCF of 144, 180 and 192 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(144, 180, 192) = HCF(HCF(144, 180), 192)

HCF(180, 144) = HCF(144, 180 mod 144) = HCF(144, 36)
HCF(144, 36) = HCF(36, 144 mod 36) = HCF(36, 0)
HCF(36, 0) = 36 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Steps for HCF(36, 192)

HCF(192, 36) = HCF(36, 192 mod 36) = HCF(36, 12)
HCF(36, 12) = HCF(12, 36 mod 12) = HCF(12, 0)
HCF(12, 0) = 12 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of HCF of 144, 180 and 192 is 12.

HCF of 144, 180 and 192 by Listing Common Factors

Factors of 144: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144
Factors of 180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180
Factors of 192: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 192

There are 6 common factors of 144, 180 and 192, that are 1, 2, 3, 4, 6, and 12. Therefore, the highest common factor of 144, 180 and 192 is 12.

HCF of 144, 180 and 192 by Long Division

HCF of 144, 180 and 192 can be represented as HCF of (HCF of 144, 180) and 192. HCF(144, 180, 192) can be thus calculated by first finding HCF(144, 180) using long division and thereafter using this result with 192 to perform long division again.

Step 1: Divide 180 (larger number) by 144 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (144) by the remainder (36). Repeat this process until the remainder = 0.
⇒ HCF(144, 180) = 36.
Step 3: Now to find the HCF of 36 and 192, we will perform a long division on 192 and 36.
Step 4: For remainder = 0, divisor = 12 ⇒ HCF(36, 192) = 12

Thus, HCF(144, 180, 192) = HCF(HCF(144, 180), 192) = 12.

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HCF of 144, 180 and 192 Examples

Example 1: Calculate the HCF of 144, 180, and 192 using LCM of the given numbers.

Solution:

Prime factorization of 144, 180 and 192 is given as,
- 144 = 2 × 2 × 2 × 2 × 3 × 3
- 180 = 2 × 2 × 3 × 3 × 5
- 192 = 2 × 2 × 2 × 2 × 2 × 2 × 3
LCM(144, 180) = 720, LCM(180, 192) = 2880, LCM(192, 144) = 576, LCM(144, 180, 192) = 2880
⇒ HCF(144, 180, 192) = [(144 × 180 × 192) × LCM(144, 180, 192)]/[LCM(144, 180) × LCM (180, 192) × LCM(192, 144)]
⇒ HCF(144, 180, 192) = (4976640 × 2880)/(720 × 2880 × 576)
⇒ HCF(144, 180, 192) = 12.
Therefore, the HCF of 144, 180 and 192 is 12.
Example 2: Find the highest number that divides 144, 180, and 192 completely.

Solution:

The highest number that divides 144, 180, and 192 exactly is their highest common factor.
- Factors of 144 = 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144
- Factors of 180 = 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180
- Factors of 192 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 192
The HCF of 144, 180, and 192 is 12.
∴ The highest number that divides 144, 180, and 192 is 12.
Example 3: Verify the relation between the LCM and HCF of 144, 180 and 192.

Solution:

The relation between the LCM and HCF of 144, 180 and 192 is given as, HCF(144, 180, 192) = [(144 × 180 × 192) × LCM(144, 180, 192)]/[LCM(144, 180) × LCM (180, 192) × LCM(144, 192)]
⇒ Prime factorization of 144, 180 and 192:
- 144 = 2 × 2 × 2 × 2 × 3 × 3
- 180 = 2 × 2 × 3 × 3 × 5
- 192 = 2 × 2 × 2 × 2 × 2 × 2 × 3
∴ LCM of (144, 180), (180, 192), (144, 192), and (144, 180, 192) is 720, 2880, 576, and 2880 respectively.
Now, LHS = HCF(144, 180, 192) = 12.
And, RHS = [(144 × 180 × 192) × LCM(144, 180, 192)]/[LCM(144, 180) × LCM (180, 192) × LCM(144, 192)] = [(4976640) × 2880]/[720 × 2880 × 576]
LHS = RHS = 12.
Hence verified.

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FAQs on HCF of 144, 180 and 192

What is the HCF of 144, 180 and 192?

The HCF of 144, 180 and 192 is 12. To calculate the highest common factor of 144, 180 and 192, we need to factor each number (factors of 144 = 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144; factors of 180 = 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180; factors of 192 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 192) and choose the highest factor that exactly divides 144, 180 and 192, i.e., 12.

Which of the following is HCF of 144, 180 and 192? 12, 209, 223, 235, 213, 218, 213

HCF of 144, 180, 192 will be the number that divides 144, 180, and 192 without leaving any remainder. The only number that satisfies the given condition is 12.

What is the Relation Between LCM and HCF of 144, 180 and 192?

The following equation can be used to express the relation between LCM and HCF of 144, 180 and 192, i.e. HCF(144, 180, 192) = [(144 × 180 × 192) × LCM(144, 180, 192)]/[LCM(144, 180) × LCM (180, 192) × LCM(144, 192)].
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How to Find the HCF of 144, 180 and 192 by Prime Factorization?

To find the HCF of 144, 180 and 192, we will find the prime factorization of given numbers, i.e. 144 = 2 × 2 × 2 × 2 × 3 × 3; 180 = 2 × 2 × 3 × 3 × 5; 192 = 2 × 2 × 2 × 2 × 2 × 2 × 3.
⇒ Since 2, 2, 3 are common terms in the prime factorization of 144, 180 and 192. Hence, HCF(144, 180, 192) = 2 × 2 × 3 = 12
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What are the Methods to Find HCF of 144, 180 and 192?

There are three commonly used methods to find the HCF of 144, 180 and 192.

By Listing Common Factors
By Long Division
By Prime Factorization

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