Find the HCF of the number in each of the following using the prime factorization method 84, 98

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HCF of 84 and 98 is the largest possible number that divides 84 and 98 exactly without any remainder. The factors of 84 and 98 are 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84 and 1, 2, 7, 14, 49, 98 respectively. There are 3 commonly used methods to find the HCF of 84 and 98 - Euclidean algorithm, long division, and prime factorization.

What is HCF of 84 and 98?

Answer: HCF of 84 and 98 is 14.

Explanation:

The HCF of two non-zero integers, x(84) and y(98), is the highest positive integer m(14) that divides both x(84) and y(98) without any remainder.

Methods to Find HCF of 84 and 98

Let's look at the different methods for finding the HCF of 84 and 98.

• Using Euclid's Algorithm
• Listing Common Factors
• Long Division Method

HCF of 84 and 98 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 = 98 and Y = 84

• HCF(98, 84) = HCF(84, 98 mod 84) = HCF(84, 14)
• HCF(84, 14) = HCF(14, 84 mod 14) = HCF(14, 0)
• HCF(14, 0) = 14 (∵ HCF(X, 0) = |X|, where X ≠ 0)

Therefore, the value of HCF of 84 and 98 is 14.

HCF of 84 and 98 by Listing Common Factors

• Factors of 84: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84
• Factors of 98: 1, 2, 7, 14, 49, 98

There are 4 common factors of 84 and 98, that are 1, 2, 14, and 7. Therefore, the highest common factor of 84 and 98 is 14.

HCF of 84 and 98 by Long Division

HCF of 84 and 98 is the divisor that we get when the remainder becomes 0 after doing long division repeatedly.

• Step 1: Divide 98 (larger number) by 84 (smaller number).
• Step 2: Since the remainder ≠ 0, we will divide the divisor of step 1 (84) by the remainder (14).
• Step 3: Repeat this process until the remainder = 0.

The corresponding divisor (14) is the HCF of 84 and 98.

☛ Also Check:

HCF of 84 and 98 Examples

1. Example 1: Find the HCF of 84 and 98, if their LCM is 588.

   Solution:

   ∵ LCM × HCF = 84 × 98 ⇒ HCF(84, 98) = (84 × 98)/588 = 14

   Therefore, the highest common factor of 84 and 98 is 14.

Show Solution >

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The HCF of 84 and 98 is 14. To calculate the Highest common factor of 84 and 98, we need to factor each number (factors of 84 = 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84; factors of 98 = 1, 2, 7, 14, 49, 98) and choose the highest factor that exactly divides both 84 and 98, i.e., 14.

If the HCF of 98 and 84 is 14, Find its LCM.

HCF(98, 84) × LCM(98, 84) = 98 × 84 Since the HCF of 98 and 84 = 14 ⇒ 14 × LCM(98, 84) = 8232 Therefore, LCM = 588

☛ Highest Common Factor Calculator

What are the Methods to Find HCF of 84 and 98?

There are three commonly used methods to find the HCF of 84 and 98.

• By Euclidean Algorithm
• By Prime Factorization
• By Long Division

How to Find the HCF of 84 and 98 by Prime Factorization?

To find the HCF of 84 and 98, we will find the prime factorization of the given numbers, i.e. 84 = 2 × 2 × 3 × 7; 98 = 2 × 7 × 7. ⇒ Since 2, 7 are common terms in the prime factorization of 84 and 98. Hence, HCF(84, 98) = 2 × 7 = 14

☛ What are Prime Numbers?

How to Find the HCF of 84 and 98 by Long Division Method?

To find the HCF of 84, 98 using long division method, 98 is divided by 84. The corresponding divisor (14) when remainder equals 0 is taken as HCF.

What is the Relation Between LCM and HCF of 84, 98?

The following equation can be used to express the relation between Least Common Multiple (LCM) and HCF of 84 and 98, i.e. HCF × LCM = 84 × 98.