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HCF of 12, 24 and 36

HCF of 12, 24 and 36 is the largest possible number that divides 12, 24 and 36 exactly without any remainder. The factors of 12, 24 and 36 are (1, 2, 3, 4, 6, 12), (1, 2, 3, 4, 6, 8, 12, 24) and (1, 2, 3, 4, 6, 9, 12, 18, 36) respectively. There are 3 commonly used methods to find the HCF of 12, 24 and 36 - Euclidean algorithm, prime factorization, and long division.

1.	HCF of 12, 24 and 36
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is HCF of 12, 24 and 36?

Answer: HCF of 12, 24 and 36 is 12.

Explanation:

The HCF of three non-zero integers, x(12), y(24) and z(36), is the highest positive integer m(12) that divides x(12), y(24) and z(36) without any remainder.

Methods to Find HCF of 12, 24 and 36

Let's look at the different methods for finding the HCF of 12, 24 and 36.

Using Euclid's Algorithm
Listing Common Factors
Prime Factorization Method

HCF of 12, 24 and 36 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(12, 24, 36) = HCF(HCF(12, 24), 36)

Steps for HCF(12, 24):

HCF(24, 12) = HCF(12, 24 mod 12) = HCF(12, 0)
HCF(12, 0) = 12 (∵ HCF(X, 0) = |X|, where X ≠ 0)

⇒ HCF(12, 24) = 12

⇒ HCF(HCF(12, 24), 36) = HCF(12, 36)

Steps for HCF(12, 36):

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

⇒ HCF(12, 36) = 12

Therefore, the value of HCF of 12, 24, and 36 is 12.

HCF of 12, 24 and 36 by Listing Common Factors

Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

There are 6 common factors of 12, 24 and 36, that are 1, 2, 3, 4, 6, and 12. Therefore, the highest common factor of 12, 24 and 36 is 12.

HCF of 12, 24 and 36 by Prime Factorization

Prime factorization of 12, 24 and 36 is (2 × 2 × 3), (2 × 2 × 2 × 3) and (2 × 2 × 3 × 3) respectively. As visible, 12, 24 and 36 have common prime factors. Hence, the HCF of 12, 24 and 36 is 2 × 2 × 3 = 12.

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HCF of 12, 24 and 36 Examples

Example 1: Calculate the HCF of 12, 24, and 36 using LCM of the given numbers.

Solution:

Prime factorization of 12, 24 and 36 is given as,
- 12 = 2 × 2 × 3
- 24 = 2 × 2 × 2 × 3
- 36 = 2 × 2 × 3 × 3
LCM(12, 24) = 24, LCM(24, 36) = 72, LCM(36, 12) = 36, LCM(12, 24, 36) = 72
⇒ HCF(12, 24, 36) = [(12 × 24 × 36) × LCM(12, 24, 36)]/[LCM(12, 24) × LCM (24, 36) × LCM(36, 12)]
⇒ HCF(12, 24, 36) = (10368 × 72)/(24 × 72 × 36)
⇒ HCF(12, 24, 36) = 12.
Therefore, the HCF of 12, 24 and 36 is 12.
Example 2: Find the highest number that divides 12, 24, and 36 completely.

Solution:

The highest number that divides 12, 24, and 36 exactly is their highest common factor.
- Factors of 12 = 1, 2, 3, 4, 6, 12
- Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24
- Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36
The HCF of 12, 24, and 36 is 12.
∴ The highest number that divides 12, 24, and 36 is 12.
Example 3: Verify the relation between the LCM and HCF of 12, 24 and 36.

Solution:

The relation between the LCM and HCF of 12, 24 and 36 is given as, HCF(12, 24, 36) = [(12 × 24 × 36) × LCM(12, 24, 36)]/[LCM(12, 24) × LCM (24, 36) × LCM(12, 36)]
⇒ Prime factorization of 12, 24 and 36:
- 12 = 2 × 2 × 3
- 24 = 2 × 2 × 2 × 3
- 36 = 2 × 2 × 3 × 3
∴ LCM of (12, 24), (24, 36), (12, 36), and (12, 24, 36) is 24, 72, 36, and 72 respectively.
Now, LHS = HCF(12, 24, 36) = 12.
And, RHS = [(12 × 24 × 36) × LCM(12, 24, 36)]/[LCM(12, 24) × LCM (24, 36) × LCM(12, 36)] = [(10368) × 72]/[24 × 72 × 36]
LHS = RHS = 12.
Hence verified.

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FAQs on HCF of 12, 24 and 36

What is the HCF of 12, 24 and 36?

The HCF of 12, 24 and 36 is 12. To calculate the HCF of 12, 24 and 36, we need to factor each number (factors of 12 = 1, 2, 3, 4, 6, 12; factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24; factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36) and choose the highest factor that exactly divides 12, 24 and 36, i.e., 12.

What is the Relation Between LCM and HCF of 12, 24 and 36?

The following equation can be used to express the relation between Least Common Multiple (LCM) and HCF of 12, 24 and 36, i.e. HCF(12, 24, 36) = [(12 × 24 × 36) × LCM(12, 24, 36)]/[LCM(12, 24) × LCM (24, 36) × LCM(12, 36)].
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What are the Methods to Find HCF of 12, 24 and 36?

There are three commonly used methods to find the HCF of 12, 24 and 36.

By Prime Factorization
By Long Division
By Euclidean Algorithm

Which of the following is HCF of 12, 24 and 36? 12, 42, 42, 61, 42, 39, 59, 67

HCF of 12, 24, 36 will be the number that divides 12, 24, and 36 without leaving any remainder. The only number that satisfies the given condition is 12.

How to Find the HCF of 12, 24 and 36 by Prime Factorization?

To find the HCF of 12, 24 and 36, we will find the prime factorization of given numbers, i.e. 12 = 2 × 2 × 3; 24 = 2 × 2 × 2 × 3; 36 = 2 × 2 × 3 × 3.
⇒ Since 2, 2, 3 are common terms in the prime factorization of 12, 24 and 36. Hence, HCF(12, 24, 36) = 2 × 2 × 3 = 12
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