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HCF of 91, 112 and 49

HCF of 91, 112 and 49 is the largest possible number that divides 91, 112 and 49 exactly without any remainder. The factors of 91, 112 and 49 are (1, 7, 13, 91), (1, 2, 4, 7, 8, 14, 16, 28, 56, 112) and (1, 7, 49) respectively. There are 3 commonly used methods to find the HCF of 91, 112 and 49 - Euclidean algorithm, prime factorization, and long division.

1.	HCF of 91, 112 and 49
2.	List of Methods
3.	Solved Examples
4.	FAQs

What is HCF of 91, 112 and 49?

Answer: HCF of 91, 112 and 49 is 7.

Explanation:

The HCF of three non-zero integers, x(91), y(112) and z(49), is the highest positive integer m(7) that divides x(91), y(112) and z(49) without any remainder.

Methods to Find HCF of 91, 112 and 49

The methods to find the HCF of 91, 112 and 49 are explained below.

Prime Factorization Method
Listing Common Factors
Using Euclid's Algorithm

HCF of 91, 112 and 49 by Prime Factorization

Prime factorization of 91, 112 and 49 is (7 × 13), (2 × 2 × 2 × 2 × 7) and (7 × 7) respectively. As visible, 91, 112 and 49 have only one common prime factor i.e. 7. Hence, the HCF of 91, 112 and 49 is 7.

HCF of 91, 112 and 49 by Listing Common Factors

Factors of 91: 1, 7, 13, 91
Factors of 112: 1, 2, 4, 7, 8, 14, 16, 28, 56, 112
Factors of 49: 1, 7, 49

There are 2 common factors of 91, 112 and 49, that are 1 and 7. Therefore, the highest common factor of 91, 112 and 49 is 7.

HCF of 91, 112 and 49 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(91, 112, 49) = HCF(HCF(91, 112), 49)

HCF(112, 91) = HCF(91, 112 mod 91) = HCF(91, 21)
HCF(91, 21) = HCF(21, 91 mod 21) = HCF(21, 7)
HCF(21, 7) = HCF(7, 21 mod 7) = HCF(7, 0)
HCF(7, 0) = 7 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Steps for HCF(7, 49)

HCF(49, 7) = HCF(7, 49 mod 7) = HCF(7, 0)
HCF(7, 0) = 7 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of HCF of 91, 112 and 49 is 7.

☛ Also Check:

HCF of 91, 112 and 49 Examples

Example 1: Find the highest number that divides 91, 112, and 49 completely.

Solution:

The highest number that divides 91, 112, and 49 exactly is their highest common factor.
- Factors of 91 = 1, 7, 13, 91
- Factors of 112 = 1, 2, 4, 7, 8, 14, 16, 28, 56, 112
- Factors of 49 = 1, 7, 49
The HCF of 91, 112, and 49 is 7.
∴ The highest number that divides 91, 112, and 49 is 7.
Example 2: Verify the relation between the LCM and HCF of 91, 112 and 49.

Solution:

The relation between the LCM and HCF of 91, 112 and 49 is given as, HCF(91, 112, 49) = [(91 × 112 × 49) × LCM(91, 112, 49)]/[LCM(91, 112) × LCM (112, 49) × LCM(91, 49)]
⇒ Prime factorization of 91, 112 and 49:
- 91 = 7 × 13
- 112 = 2 × 2 × 2 × 2 × 7
- 49 = 7 × 7
∴ LCM of (91, 112), (112, 49), (91, 49), and (91, 112, 49) is 1456, 784, 637, and 10192 respectively.
Now, LHS = HCF(91, 112, 49) = 7.
And, RHS = [(91 × 112 × 49) × LCM(91, 112, 49)]/[LCM(91, 112) × LCM (112, 49) × LCM(91, 49)] = [(499408) × 10192]/[1456 × 784 × 637]
LHS = RHS = 7.
Hence verified.
Example 3: Calculate the HCF of 91, 112, and 49 using LCM of the given numbers.

Solution:

Prime factorization of 91, 112 and 49 is given as,
- 91 = 7 × 13
- 112 = 2 × 2 × 2 × 2 × 7
- 49 = 7 × 7
LCM(91, 112) = 1456, LCM(112, 49) = 784, LCM(49, 91) = 637, LCM(91, 112, 49) = 10192
⇒ HCF(91, 112, 49) = [(91 × 112 × 49) × LCM(91, 112, 49)]/[LCM(91, 112) × LCM (112, 49) × LCM(49, 91)]
⇒ HCF(91, 112, 49) = (499408 × 10192)/(1456 × 784 × 637)
⇒ HCF(91, 112, 49) = 7.
Therefore, the HCF of 91, 112 and 49 is 7.

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FAQs on HCF of 91, 112 and 49

What is the HCF of 91, 112 and 49?

The HCF of 91, 112 and 49 is 7. To calculate the highest common factor (HCF) of 91, 112 and 49, we need to factor each number (factors of 91 = 1, 7, 13, 91; factors of 112 = 1, 2, 4, 7, 8, 14, 16, 28, 56, 112; factors of 49 = 1, 7, 49) and choose the highest factor that exactly divides 91, 112 and 49, i.e., 7.

What is the Relation Between LCM and HCF of 91, 112 and 49?

The following equation can be used to express the relation between Least Common Multiple (LCM) and HCF of 91, 112 and 49, i.e. HCF(91, 112, 49) = [(91 × 112 × 49) × LCM(91, 112, 49)]/[LCM(91, 112) × LCM (112, 49) × LCM(91, 49)].
☛ HCF Calculator

How to Find the HCF of 91, 112 and 49 by Prime Factorization?

To find the HCF of 91, 112 and 49, we will find the prime factorization of given numbers, i.e. 91 = 7 × 13; 112 = 2 × 2 × 2 × 2 × 7; 49 = 7 × 7.
⇒ Since 7 is the only common prime factor of 91, 112 and 49. Hence, HCF(91, 112, 49) = 7.
☛ Prime Number

Which of the following is HCF of 91, 112 and 49? 7, 157, 126, 132, 114, 158, 133

HCF of 91, 112, 49 will be the number that divides 91, 112, and 49 without leaving any remainder. The only number that satisfies the given condition is 7.

What are the Methods to Find HCF of 91, 112 and 49?

There are three commonly used methods to find the HCF of 91, 112 and 49.

By Long Division
By Listing Common Factors
By Prime Factorization

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