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HCF of 12, 15 and 21

HCF of 12, 15 and 21 is the largest possible number that divides 12, 15 and 21 exactly without any remainder. The factors of 12, 15 and 21 are (1, 2, 3, 4, 6, 12), (1, 3, 5, 15) and (1, 3, 7, 21) respectively. There are 3 commonly used methods to find the HCF of 12, 15 and 21 - Euclidean algorithm, prime factorization, and long division.

1.	HCF of 12, 15 and 21
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is HCF of 12, 15 and 21?

Answer: HCF of 12, 15 and 21 is 3.

Explanation:

The HCF of three non-zero integers, x(12), y(15) and z(21), is the highest positive integer m(3) that divides x(12), y(15) and z(21) without any remainder.

Methods to Find HCF of 12, 15 and 21

Let's look at the different methods for finding the HCF of 12, 15 and 21.

Long Division Method
Using Euclid's Algorithm
Prime Factorization Method

HCF of 12, 15 and 21 by Long Division

HCF of 12, 15 and 21 can be represented as HCF of (HCF of 12, 15) and 21. HCF(12, 15, 21) can be thus calculated by first finding HCF(12, 15) using long division and thereafter using this result with 21 to perform long division again.

Step 1: Divide 15 (larger number) by 12 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (12) by the remainder (3). Repeat this process until the remainder = 0.
⇒ HCF(12, 15) = 3.
Step 3: Now to find the HCF of 3 and 21, we will perform a long division on 21 and 3.
Step 4: For remainder = 0, divisor = 3 ⇒ HCF(3, 21) = 3

Thus, HCF(12, 15, 21) = HCF(HCF(12, 15), 21) = 3.

HCF of 12, 15 and 21 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, 15, 21) = HCF(HCF(12, 15), 21)

HCF(15, 12) = HCF(12, 15 mod 12) = HCF(12, 3)
HCF(12, 3) = HCF(3, 12 mod 3) = HCF(3, 0)
HCF(3, 0) = 3 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Steps for HCF(3, 21)

HCF(21, 3) = HCF(3, 21 mod 3) = HCF(3, 0)
HCF(3, 0) = 3 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of HCF of 12, 15 and 21 is 3.

HCF of 12, 15 and 21 by Prime Factorization

Prime factorization of 12, 15 and 21 is (2 × 2 × 3), (3 × 5) and (3 × 7) respectively. As visible, 12, 15 and 21 have only one common prime factor i.e. 3. Hence, the HCF of 12, 15 and 21 is 3.

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HCF of 12, 15 and 21 Examples

Example 1: Verify the relation between the LCM and HCF of 12, 15 and 21.

Solution:

The relation between the LCM and HCF of 12, 15 and 21 is given as, HCF(12, 15, 21) = [(12 × 15 × 21) × LCM(12, 15, 21)]/[LCM(12, 15) × LCM (15, 21) × LCM(12, 21)]
⇒ Prime factorization of 12, 15 and 21:
- 12 = 2 × 2 × 3
- 15 = 3 × 5
- 21 = 3 × 7
∴ LCM of (12, 15), (15, 21), (12, 21), and (12, 15, 21) is 60, 105, 84, and 420 respectively.
Now, LHS = HCF(12, 15, 21) = 3.
And, RHS = [(12 × 15 × 21) × LCM(12, 15, 21)]/[LCM(12, 15) × LCM (15, 21) × LCM(12, 21)] = [(3780) × 420]/[60 × 105 × 84]
LHS = RHS = 3.
Hence verified.
Example 2: Calculate the HCF of 12, 15, and 21 using LCM of the given numbers.

Solution:

Prime factorization of 12, 15 and 21 is given as,
- 12 = 2 × 2 × 3
- 15 = 3 × 5
- 21 = 3 × 7
LCM(12, 15) = 60, LCM(15, 21) = 105, LCM(21, 12) = 84, LCM(12, 15, 21) = 420
⇒ HCF(12, 15, 21) = [(12 × 15 × 21) × LCM(12, 15, 21)]/[LCM(12, 15) × LCM (15, 21) × LCM(21, 12)]
⇒ HCF(12, 15, 21) = (3780 × 420)/(60 × 105 × 84)
⇒ HCF(12, 15, 21) = 3.
Therefore, the HCF of 12, 15 and 21 is 3.
Example 3: Find the highest number that divides 12, 15, and 21 completely.

Solution:

The highest number that divides 12, 15, and 21 exactly is their highest common factor.
- Factors of 12 = 1, 2, 3, 4, 6, 12
- Factors of 15 = 1, 3, 5, 15
- Factors of 21 = 1, 3, 7, 21
The HCF of 12, 15, and 21 is 3.
∴ The highest number that divides 12, 15, and 21 is 3.

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FAQs on HCF of 12, 15 and 21

What is the HCF of 12, 15 and 21?

The HCF of 12, 15 and 21 is 3. To calculate the highest common factor (HCF) of 12, 15 and 21, we need to factor each number (factors of 12 = 1, 2, 3, 4, 6, 12; factors of 15 = 1, 3, 5, 15; factors of 21 = 1, 3, 7, 21) and choose the highest factor that exactly divides 12, 15 and 21, i.e., 3.

How to Find the HCF of 12, 15 and 21 by Prime Factorization?

To find the HCF of 12, 15 and 21, we will find the prime factorization of given numbers, i.e. 12 = 2 × 2 × 3; 15 = 3 × 5; 21 = 3 × 7.
⇒ Since 3 is the only common prime factor of 12, 15 and 21. Hence, HCF(12, 15, 21) = 3.
☛ What is a Prime Number?

What is the Relation Between LCM and HCF of 12, 15 and 21?

The following equation can be used to express the relation between LCM and HCF of 12, 15 and 21, i.e. HCF(12, 15, 21) = [(12 × 15 × 21) × LCM(12, 15, 21)]/[LCM(12, 15) × LCM (15, 21) × LCM(12, 21)].
☛ HCF Calculator

What are the Methods to Find HCF of 12, 15 and 21?

There are three commonly used methods to find the HCF of 12, 15 and 21.

By Prime Factorization
By Euclidean Algorithm
By Long Division

Which of the following is HCF of 12, 15 and 21? 3, 47, 36, 57, 69

HCF of 12, 15, 21 will be the number that divides 12, 15, and 21 without leaving any remainder. The only number that satisfies the given condition is 3.

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