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HCF of 40, 42 and 45

HCF of 40, 42 and 45 is the largest possible number that divides 40, 42 and 45 exactly without any remainder. The factors of 40, 42 and 45 are (1, 2, 4, 5, 8, 10, 20, 40), (1, 2, 3, 6, 7, 14, 21, 42) and (1, 3, 5, 9, 15, 45) respectively. There are 3 commonly used methods to find the HCF of 40, 42 and 45 - Euclidean algorithm, long division, and prime factorization.

1.	HCF of 40, 42 and 45
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is HCF of 40, 42 and 45?

Answer: HCF of 40, 42 and 45 is 1.

Explanation:

The HCF of three non-zero integers, x(40), y(42) and z(45), is the highest positive integer m(1) that divides x(40), y(42) and z(45) without any remainder.

Methods to Find HCF of 40, 42 and 45

The methods to find the HCF of 40, 42 and 45 are explained below.

Prime Factorization Method
Listing Common Factors
Using Euclid's Algorithm

HCF of 40, 42 and 45 by Prime Factorization

Prime factorization of 40, 42 and 45 is (2 × 2 × 2 × 5), (2 × 3 × 7) and (3 × 3 × 5) respectively. As visible, there are no common prime factors between 40, 42 and 45, i.e. they are coprime. Hence, the HCF of 40, 42 and 45 will be 1.

HCF of 40, 42 and 45 by Listing Common Factors

Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
Factors of 45: 1, 3, 5, 9, 15, 45

Since, 1 is the only common factor between 40, 42 and 45. The Highest Common Factor of 40, 42 and 45 is 1.

HCF of 40, 42 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.

HCF(40, 42, 45) = HCF(HCF(40, 42), 45)

HCF(42, 40) = HCF(40, 42 mod 40) = HCF(40, 2)
HCF(40, 2) = HCF(2, 40 mod 2) = HCF(2, 0)
HCF(2, 0) = 2 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Steps for HCF(2, 45)

HCF(45, 2) = HCF(2, 45 mod 2) = HCF(2, 1)
HCF(2, 1) = HCF(1, 2 mod 1) = HCF(1, 0)
HCF(1, 0) = 1 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of HCF of 40, 42 and 45 is 1.

☛ Also Check:

HCF of 40, 42 and 45 Examples

Example 1: Verify the relation between the LCM and HCF of 40, 42 and 45.

Solution:

The relation between the LCM and HCF of 40, 42 and 45 is given as, HCF(40, 42, 45) = [(40 × 42 × 45) × LCM(40, 42, 45)]/[LCM(40, 42) × LCM (42, 45) × LCM(40, 45)]
⇒ Prime factorization of 40, 42 and 45:
- 40 = 2 × 2 × 2 × 5
- 42 = 2 × 3 × 7
- 45 = 3 × 3 × 5
∴ LCM of (40, 42), (42, 45), (40, 45), and (40, 42, 45) is 840, 630, 360, and 2520 respectively.
Now, LHS = HCF(40, 42, 45) = 1.
And, RHS = [(40 × 42 × 45) × LCM(40, 42, 45)]/[LCM(40, 42) × LCM (42, 45) × LCM(40, 45)] = [(75600) × 2520]/[840 × 630 × 360]
LHS = RHS = 1.
Hence verified.
Example 2: Calculate the HCF of 40, 42, and 45 using LCM of the given numbers.

Solution:

Prime factorization of 40, 42 and 45 is given as,
- 40 = 2 × 2 × 2 × 5
- 42 = 2 × 3 × 7
- 45 = 3 × 3 × 5
LCM(40, 42) = 840, LCM(42, 45) = 630, LCM(45, 40) = 360, LCM(40, 42, 45) = 2520
⇒ HCF(40, 42, 45) = [(40 × 42 × 45) × LCM(40, 42, 45)]/[LCM(40, 42) × LCM (42, 45) × LCM(45, 40)]
⇒ HCF(40, 42, 45) = (75600 × 2520)/(840 × 630 × 360)
⇒ HCF(40, 42, 45) = 1.
Therefore, the HCF of 40, 42 and 45 is 1.
Example 3: Find the highest number that divides 40, 42, and 45 completely.

Solution:

The highest number that divides 40, 42, and 45 exactly is their highest common factor.
- Factors of 40 = 1, 2, 4, 5, 8, 10, 20, 40
- Factors of 42 = 1, 2, 3, 6, 7, 14, 21, 42
- Factors of 45 = 1, 3, 5, 9, 15, 45
The HCF of 40, 42, and 45 is 1.
∴ The highest number that divides 40, 42, and 45 is 1.

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FAQs on HCF of 40, 42 and 45

What is the HCF of 40, 42 and 45?

The HCF of 40, 42 and 45 is 1. To calculate the highest common factor of 40, 42 and 45, we need to factor each number (factors of 40 = 1, 2, 4, 5, 8, 10, 20, 40; factors of 42 = 1, 2, 3, 6, 7, 14, 21, 42; factors of 45 = 1, 3, 5, 9, 15, 45) and choose the highest factor that exactly divides 40, 42 and 45, i.e., 1.

How to Find the HCF of 40, 42 and 45 by Prime Factorization?

To find the HCF of 40, 42 and 45, we will find the prime factorization of given numbers, i.e. 40 = 2 × 2 × 2 × 5; 42 = 2 × 3 × 7; 45 = 3 × 3 × 5.
⇒ There is no common prime factor for 40, 42 and 45. Hence, HCF(40, 42, 45) = 1.
☛ What is a Prime Number?

What are the Methods to Find HCF of 40, 42 and 45?

There are three commonly used methods to find the HCF of 40, 42 and 45.

By Listing Common Factors
By Long Division
By Prime Factorization

What is the Relation Between LCM and HCF of 40, 42 and 45?

The following equation can be used to express the relation between LCM (Least Common Multiple) and HCF of 40, 42 and 45, i.e. HCF(40, 42, 45) = [(40 × 42 × 45) × LCM(40, 42, 45)]/[LCM(40, 42) × LCM (42, 45) × LCM(40, 45)].
☛ Highest Common Factor Calculator

Which of the following is HCF of 40, 42 and 45? 1, 49, 71, 45, 89, 67, 49

HCF of 40, 42, 45 will be the number that divides 40, 42, and 45 without leaving any remainder. The only number that satisfies the given condition is 1.

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