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HCF of 27 and 45

HCF of 27 and 45 is the largest possible number that divides 27 and 45 exactly without any remainder. The factors of 27 and 45 are 1, 3, 9, 27 and 1, 3, 5, 9, 15, 45 respectively. There are 3 commonly used methods to find the HCF of 27 and 45 - prime factorization, long division, and Euclidean algorithm.

1.	HCF of 27 and 45
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is HCF of 27 and 45?

Answer: HCF of 27 and 45 is 9.

Explanation:

The HCF of two non-zero integers, x(27) and y(45), is the highest positive integer m(9) that divides both x(27) and y(45) without any remainder.

Methods to Find HCF of 27 and 45

The methods to find the HCF of 27 and 45 are explained below.

Listing Common Factors
Using Euclid's Algorithm
Prime Factorization Method

HCF of 27 and 45 by Listing Common Factors

Factors of 27: 1, 3, 9, 27
Factors of 45: 1, 3, 5, 9, 15, 45

There are 3 common factors of 27 and 45, that are 1, 3, and 9. Therefore, the highest common factor of 27 and 45 is 9.

HCF of 27 and 45 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 = 45 and Y = 27

HCF(45, 27) = HCF(27, 45 mod 27) = HCF(27, 18)
HCF(27, 18) = HCF(18, 27 mod 18) = HCF(18, 9)
HCF(18, 9) = HCF(9, 18 mod 9) = HCF(9, 0)
HCF(9, 0) = 9 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of HCF of 27 and 45 is 9.

HCF of 27 and 45 by Prime Factorization

Prime factorization of 27 and 45 is (3 × 3 × 3) and (3 × 3 × 5) respectively. As visible, 27 and 45 have common prime factors. Hence, the HCF of 27 and 45 is 3 × 3 = 9.

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HCF of 27 and 45 Examples

Example 1: For two numbers, HCF = 9 and LCM = 135. If one number is 45, find the other number.

Solution:

Given: HCF (z, 45) = 9 and LCM (z, 45) = 135
∵ HCF × LCM = 45 × (z)
⇒ z = (HCF × LCM)/45
⇒ z = (9 × 135)/45
⇒ z = 27
Therefore, the other number is 27.
Example 2: Find the HCF of 27 and 45, if their LCM is 135.

Solution:

∵ LCM × HCF = 27 × 45
⇒ HCF(27, 45) = (27 × 45)/135 = 9
Therefore, the highest common factor of 27 and 45 is 9.
Example 3: Find the highest number that divides 27 and 45 exactly.

Solution:

The highest number that divides 27 and 45 exactly is their highest common factor, i.e. HCF of 27 and 45.
⇒ Factors of 27 and 45:
- Factors of 27 = 1, 3, 9, 27
- Factors of 45 = 1, 3, 5, 9, 15, 45
Therefore, the HCF of 27 and 45 is 9.

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FAQs on HCF of 27 and 45

What is the HCF of 27 and 45?

The HCF of 27 and 45 is 9. To calculate the HCF of 27 and 45, we need to factor each number (factors of 27 = 1, 3, 9, 27; factors of 45 = 1, 3, 5, 9, 15, 45) and choose the highest factor that exactly divides both 27 and 45, i.e., 9.

How to Find the HCF of 27 and 45 by Prime Factorization?

To find the HCF of 27 and 45, we will find the prime factorization of the given numbers, i.e. 27 = 3 × 3 × 3; 45 = 3 × 3 × 5.
⇒ Since 3, 3 are common terms in the prime factorization of 27 and 45. Hence, HCF(27, 45) = 3 × 3 = 9
☛ Prime Numbers

What is the Relation Between LCM and HCF of 27, 45?

The following equation can be used to express the relation between LCM (Least Common Multiple) and HCF of 27 and 45, i.e. HCF × LCM = 27 × 45.

If the HCF of 45 and 27 is 9, Find its LCM.

HCF(45, 27) × LCM(45, 27) = 45 × 27
Since the HCF of 45 and 27 = 9
⇒ 9 × LCM(45, 27) = 1215
Therefore, LCM = 135
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How to Find the HCF of 27 and 45 by Long Division Method?

To find the HCF of 27, 45 using long division method, 45 is divided by 27. The corresponding divisor (9) when remainder equals 0 is taken as HCF.

What are the Methods to Find HCF of 27 and 45?

There are three commonly used methods to find the HCF of 27 and 45.

By Long Division
By Euclidean Algorithm
By Prime Factorization

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