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HCF of 12 and 15

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

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

What is HCF of 12 and 15?

Answer: HCF of 12 and 15 is 3.

Explanation:

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

Methods to Find HCF of 12 and 15

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

Prime Factorization Method
Listing Common Factors
Long Division Method

HCF of 12 and 15 by Prime Factorization

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

HCF of 12 and 15 by Listing Common Factors

Factors of 12: 1, 2, 3, 4, 6, 12
Factors of 15: 1, 3, 5, 15

There are 2 common factors of 12 and 15, that are 1 and 3. Therefore, the highest common factor of 12 and 15 is 3.

HCF of 12 and 15 by Long Division

HCF of 12 and 15 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

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).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (3) is the HCF of 12 and 15.

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HCF of 12 and 15 Examples

Example 1: Find the HCF of 12 and 15, if their LCM is 60.

Solution:

∵ LCM × HCF = 12 × 15
⇒ HCF(12, 15) = (12 × 15)/60 = 3
Therefore, the highest common factor of 12 and 15 is 3.
Example 2: For two numbers, HCF = 3 and LCM = 60. If one number is 12, find the other number.

Solution:

Given: HCF (y, 12) = 3 and LCM (y, 12) = 60
∵ HCF × LCM = 12 × (y)
⇒ y = (HCF × LCM)/12
⇒ y = (3 × 60)/12
⇒ y = 15
Therefore, the other number is 15.
Example 3: Find the highest number that divides 12 and 15 exactly.

Solution:

The highest number that divides 12 and 15 exactly is their highest common factor, i.e. HCF of 12 and 15.
⇒ Factors of 12 and 15:
- Factors of 12 = 1, 2, 3, 4, 6, 12
- Factors of 15 = 1, 3, 5, 15
Therefore, the HCF of 12 and 15 is 3.

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FAQs on HCF of 12 and 15

What is the HCF of 12 and 15?

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

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

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

How to Find the HCF of 12 and 15 by Long Division Method?

To find the HCF of 12, 15 using long division method, 15 is divided by 12. The corresponding divisor (3) when remainder equals 0 is taken as HCF.

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

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

By Listing Common Factors
By Prime Factorization
By Long Division

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

The following equation can be used to express the relation between Least Common Multiple (LCM) and HCF of 12 and 15, i.e. HCF × LCM = 12 × 15.

If the HCF of 15 and 12 is 3, Find its LCM.

HCF(15, 12) × LCM(15, 12) = 15 × 12
Since the HCF of 15 and 12 = 3
⇒ 3 × LCM(15, 12) = 180
Therefore, LCM = 60
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