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HCF of 9 and 21

HCF of 9 and 21 is the largest possible number that divides 9 and 21 exactly without any remainder. The factors of 9 and 21 are 1, 3, 9 and 1, 3, 7, 21 respectively. There are 3 commonly used methods to find the HCF of 9 and 21 - long division, prime factorization, and Euclidean algorithm.

1.	HCF of 9 and 21
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is HCF of 9 and 21?

Answer: HCF of 9 and 21 is 3.

Explanation:

The HCF of two non-zero integers, x(9) and y(21), is the highest positive integer m(3) that divides both x(9) and y(21) without any remainder.

Methods to Find HCF of 9 and 21

The methods to find the HCF of 9 and 21 are explained below.

Using Euclid's Algorithm
Long Division Method
Prime Factorization Method

HCF of 9 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.

Here X = 21 and Y = 9

HCF(21, 9) = HCF(9, 21 mod 9) = HCF(9, 3)
HCF(9, 3) = HCF(3, 9 mod 3) = HCF(3, 0)
HCF(3, 0) = 3 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of HCF of 9 and 21 is 3.

HCF of 9 and 21 by Long Division

HCF of 9 and 21 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

Step 1: Divide 21 (larger number) by 9 (smaller number).
Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (9) by the remainder (3).
Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (3) is the HCF of 9 and 21.

HCF of 9 and 21 by Prime Factorization

Prime factorization of 9 and 21 is (3 × 3) and (3 × 7) respectively. As visible, 9 and 21 have only one common prime factor i.e. 3. Hence, the HCF of 9 and 21 is 3.

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HCF of 9 and 21 Examples

Example 1: Find the HCF of 9 and 21, if their LCM is 63.

Solution:

∵ LCM × HCF = 9 × 21
⇒ HCF(9, 21) = (9 × 21)/63 = 3
Therefore, the highest common factor of 9 and 21 is 3.
Example 2: For two numbers, HCF = 3 and LCM = 63. If one number is 21, find the other number.

Solution:

Given: HCF (x, 21) = 3 and LCM (x, 21) = 63
∵ HCF × LCM = 21 × (x)
⇒ x = (HCF × LCM)/21
⇒ x = (3 × 63)/21
⇒ x = 9
Therefore, the other number is 9.
Example 3: Find the highest number that divides 9 and 21 exactly.

Solution:

The highest number that divides 9 and 21 exactly is their highest common factor, i.e. HCF of 9 and 21.
⇒ Factors of 9 and 21:
- Factors of 9 = 1, 3, 9
- Factors of 21 = 1, 3, 7, 21
Therefore, the HCF of 9 and 21 is 3.

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FAQs on HCF of 9 and 21

What is the HCF of 9 and 21?

The HCF of 9 and 21 is 3. To calculate the HCF (Highest Common Factor) of 9 and 21, we need to factor each number (factors of 9 = 1, 3, 9; factors of 21 = 1, 3, 7, 21) and choose the highest factor that exactly divides both 9 and 21, i.e., 3.

If the HCF of 21 and 9 is 3, Find its LCM.

HCF(21, 9) × LCM(21, 9) = 21 × 9
Since the HCF of 21 and 9 = 3
⇒ 3 × LCM(21, 9) = 189
Therefore, LCM = 63
☛ Highest Common Factor Calculator

How to Find the HCF of 9 and 21 by Long Division Method?

To find the HCF of 9, 21 using long division method, 21 is divided by 9. The corresponding divisor (3) when remainder equals 0 is taken as HCF.

How to Find the HCF of 9 and 21 by Prime Factorization?

To find the HCF of 9 and 21, we will find the prime factorization of the given numbers, i.e. 9 = 3 × 3; 21 = 3 × 7.
⇒ Since 3 is the only common prime factor of 9 and 21. Hence, HCF (9, 21) = 3.
☛ Prime Number

What is the Relation Between LCM and HCF of 9, 21?

The following equation can be used to express the relation between Least Common Multiple and HCF of 9 and 21, i.e. HCF × LCM = 9 × 21.

What are the Methods to Find HCF of 9 and 21?

There are three commonly used methods to find the HCF of 9 and 21.

By Long Division
By Euclidean Algorithm
By Prime Factorization

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