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HCF of 12, 16 and 24

HCF of 12, 16 and 24 is the largest possible number that divides 12, 16 and 24 exactly without any remainder. The factors of 12, 16 and 24 are (1, 2, 3, 4, 6, 12), (1, 2, 4, 8, 16) and (1, 2, 3, 4, 6, 8, 12, 24) respectively. There are 3 commonly used methods to find the HCF of 12, 16 and 24 - prime factorization, Euclidean algorithm, and long division.

1.	HCF of 12, 16 and 24
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is HCF of 12, 16 and 24?

Answer: HCF of 12, 16 and 24 is 4.

Explanation:

The HCF of three non-zero integers, x(12), y(16) and z(24), is the highest positive integer m(4) that divides x(12), y(16) and z(24) without any remainder.

Methods to Find HCF of 12, 16 and 24

The methods to find the HCF of 12, 16 and 24 are explained below.

Long Division Method
Listing Common Factors
Using Euclid's Algorithm

HCF of 12, 16 and 24 by Long Division

HCF of 12, 16 and 24 can be represented as HCF of (HCF of 12, 16) and 24. HCF(12, 16, 24) can be thus calculated by first finding HCF(12, 16) using long division and thereafter using this result with 24 to perform long division again.

Step 1: Divide 16 (larger number) by 12 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (12) by the remainder (4). Repeat this process until the remainder = 0.
⇒ HCF(12, 16) = 4.
Step 3: Now to find the HCF of 4 and 24, we will perform a long division on 24 and 4.
Step 4: For remainder = 0, divisor = 4 ⇒ HCF(4, 24) = 4

Thus, HCF(12, 16, 24) = HCF(HCF(12, 16), 24) = 4.

HCF of 12, 16 and 24 by Listing Common Factors

Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 16: 1, 2, 4, 8, 16
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

There are 3 common factors of 12, 16 and 24, that are 1, 2, and 4. Therefore, the highest common factor of 12, 16 and 24 is 4.

HCF of 12, 16 and 24 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, 16, 24) = HCF(HCF(12, 16), 24)

HCF(16, 12) = HCF(12, 16 mod 12) = HCF(12, 4)
HCF(12, 4) = HCF(4, 12 mod 4) = HCF(4, 0)
HCF(4, 0) = 4 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Steps for HCF(4, 24)

HCF(24, 4) = HCF(4, 24 mod 4) = HCF(4, 0)
HCF(4, 0) = 4 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of HCF of 12, 16 and 24 is 4.

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HCF of 12, 16 and 24 Examples

Example 1: Find the highest number that divides 12, 16, and 24 completely.

Solution:

The highest number that divides 12, 16, and 24 exactly is their highest common factor.
- Factors of 12 = 1, 2, 3, 4, 6, 12
- Factors of 16 = 1, 2, 4, 8, 16
- Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24
The HCF of 12, 16, and 24 is 4.
∴ The highest number that divides 12, 16, and 24 is 4.
Example 2: Verify the relation between the LCM and HCF of 12, 16 and 24.

Solution:

The relation between the LCM and HCF of 12, 16 and 24 is given as, HCF(12, 16, 24) = [(12 × 16 × 24) × LCM(12, 16, 24)]/[LCM(12, 16) × LCM (16, 24) × LCM(12, 24)]
⇒ Prime factorization of 12, 16 and 24:
- 12 = 2 × 2 × 3
- 16 = 2 × 2 × 2 × 2
- 24 = 2 × 2 × 2 × 3
∴ LCM of (12, 16), (16, 24), (12, 24), and (12, 16, 24) is 48, 48, 24, and 48 respectively.
Now, LHS = HCF(12, 16, 24) = 4.
And, RHS = [(12 × 16 × 24) × LCM(12, 16, 24)]/[LCM(12, 16) × LCM (16, 24) × LCM(12, 24)] = [(4608) × 48]/[48 × 48 × 24]
LHS = RHS = 4.
Hence verified.
Example 3: Calculate the HCF of 12, 16, and 24 using LCM of the given numbers.

Solution:

Prime factorization of 12, 16 and 24 is given as,
- 12 = 2 × 2 × 3
- 16 = 2 × 2 × 2 × 2
- 24 = 2 × 2 × 2 × 3
LCM(12, 16) = 48, LCM(16, 24) = 48, LCM(24, 12) = 24, LCM(12, 16, 24) = 48
⇒ HCF(12, 16, 24) = [(12 × 16 × 24) × LCM(12, 16, 24)]/[LCM(12, 16) × LCM (16, 24) × LCM(24, 12)]
⇒ HCF(12, 16, 24) = (4608 × 48)/(48 × 48 × 24)
⇒ HCF(12, 16, 24) = 4.
Therefore, the HCF of 12, 16 and 24 is 4.

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FAQs on HCF of 12, 16 and 24

What is the HCF of 12, 16 and 24?

The HCF of 12, 16 and 24 is 4. To calculate the HCF (Highest Common Factor) of 12, 16 and 24, we need to factor each number (factors of 12 = 1, 2, 3, 4, 6, 12; factors of 16 = 1, 2, 4, 8, 16; factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24) and choose the highest factor that exactly divides 12, 16 and 24, i.e., 4.

How to Find the HCF of 12, 16 and 24 by Prime Factorization?

To find the HCF of 12, 16 and 24, we will find the prime factorization of given numbers, i.e. 12 = 2 × 2 × 3; 16 = 2 × 2 × 2 × 2; 24 = 2 × 2 × 2 × 3.
⇒ Since 2, 2 are common terms in the prime factorization of 12, 16 and 24. Hence, HCF(12, 16, 24) = 2 × 2 = 4
☛ Prime Numbers

What are the Methods to Find HCF of 12, 16 and 24?

There are three commonly used methods to find the HCF of 12, 16 and 24.

By Long Division
By Prime Factorization
By Listing Common Factors

Which of the following is HCF of 12, 16 and 24? 4, 25, 51, 63, 62, 66, 64, 39

HCF of 12, 16, 24 will be the number that divides 12, 16, and 24 without leaving any remainder. The only number that satisfies the given condition is 4.

What is the Relation Between LCM and HCF of 12, 16 and 24?

The following equation can be used to express the relation between LCM and HCF of 12, 16 and 24, i.e. HCF(12, 16, 24) = [(12 × 16 × 24) × LCM(12, 16, 24)]/[LCM(12, 16) × LCM (16, 24) × LCM(12, 24)].
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