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# HCF Formula

The HCF formula helps in finding the highest common factor of two or more numbers. The HCF of two or more than two numbers is considered the greatest number that divides all the given numbers exactly leaving no remainder. The formula of HCF is correlated with LCM i.e. least common multiple. Let's see the various formulas and solve examples for a better understanding of the concept.

## Meaning of HCF Formula

The HCF formula helps in finding the HCF of two or more numbers. HCF is the highest or greatest common divisor of any two or more natural numbers. There are two methods of determining the HCF of a number:

- Prime Factorization Method: Two or more numbers are expressed as the product of their prime factors.
**HCF = Product of the common prime factors with the lowest powers.** - Division Method: We use Euclid's Division Lemma( a = bq+ r.) to find the HCF of two numbers a and b. i.e.
**Dividend = Divisor × Quotient + Remainder.**The lemma states when a divides b, q is the quotient and r is the remainder. If r ≠ 0, r becomes the new divisor(b) and b becomes the new dividend(a). Keep dividing until r = 0. If r = 0, then b is the HCF.

## HCF Formulas

There are different HCF formulas that are used to determine the HCF of the given numbers. Let's see what they are:

- LCM × HCF = Product of the Numbers. Suppose P and Q are two numbers, then - LCM (P & Q) × HCF (P & Q) = P × Q.
- HCF of co-prime numbers is always 1. Therefore, LCM of Co-prime Numbers = Product of the numbers
- Euclid's Division Lemma is used to find the HCF of 2 numbers. Get two numbers a and b, such that a> b. Find a/b. i.e. Dividend = Divisor × Quotient + Remainder. If the remainder is 0 then the divisor is the HCF, else apply the lemma on b and r and keep dividing until the remainder is 0.
- To find the HCF of fractions, we use HCF of fractions = HCF of numerators/LCM of the denominators
- To find the greatest number that will exactly divide p, q, and r. Required number = HCF of p, q, and r
- To find the greatest number that will divide p, q, and r leaving remainders a, b and c respectively. Required number = HCF of (p -a), (q- b) and (r – c)
- HCF of any two or more numbers is never greater than any of the given numbers.

## Examples using HCF Formulas

**Example 1: Prove the HCF formula: LCM (12 & 15) × HCF (12 & 15) = Product of 12 and 15**

**Solution:** First finding the HCF and LCM of both the numbers

12 = 2×2×3 and 15 = 3×5

LCM = 2×2×3×5 = 60 HCF = 3

The HCF formula states that LCM × HCF = Product of the Numbers

Product of the numbers = 12 × 15 = 180

LCM (12 & 15) × HCF (12 & 15) = 60 × 3 = 180

Hence, it is proved that LCM (12 & 15) × HCF (12 & 15) = Product of 12 and 15.

**Example 2: 6 and 7 are two co-prime numbers. Using these numbers verify the HCF formula, LCM of Co-prime Numbers = Product of the numbers.**

**Solution: **Let's find the LCM and HCF of 6 and 7

6 = 1×2×3

7= 1×7

LCM = 1×2×3×7 = 42 HCF = 1

The HCF formula states that LCM of Co-prime Numbers = Product of the numbers

Product of the numbers = 6×7 = 42

Hence, LCM of Co-prime Numbers = Product of the numbers i.e. 42 = 42. Therefore, it is verified.

**Example 3: Find the HCF of 9/10, 6/15, 12/20, 18/5**

**Solution:** According to the HCF formula, HCF = HCF of numerator/LCM of the denominator.

So finding the HCF of the numerators:

9 = 1×3×3

6 = 1×2×3

12 = 1×2×2×3

18 = 1×2 × 3 × 3

Therefore, the HCF of 9, 8, 12, 18 = 3

10 = 1×2×5

15 = 1×3×5

20 = 1×2×4×5

5 = 1×5

Therefore, the LCM of 10, 15, 20, 5 = 2×5×2×4×5 = 400

HCF = HCF of numerators/LCM of the denominators

HCF = 3/400

Therefore, the HCF of 9/10, 6/15, 12/20, 18/5 is 3/400

## FAQs on HCF Formula

### What is the Meaning of the HCF Formula?

The HCF formula helps in finding the HCF of numbers, LCM of numbers, and the greatest factor that divides the given numbers. The formula of the HCF helps in breaking down two or more natural numbers and finding either the HCF and LCM. HCF is the greatest common factor of any two or more numbers while LCM is the least common multiple of any two or more numbers.

### What are the Two Methods Used in HCF Formula?

To the HCF of two or more numbers, there are two methods used.

- Prime Factorization Method: The product of two or more numbers are written, the common factors or numbers are further multiplied to obtain the HCF of the number.
- Division Method: Euclid's Division Lemma is used to find the HCF of 2 numbers. Get two numbers a and b, such that a> b. Find a/b. i.e. Dividend = Divisor × Quotient + Remainder. If the remainder is 0 then the divisor is the HCF, else apply the lemma on b and r and keep dividing until the remainder is 0.

### Which HCF Formula has Two Numbers Equivalent to the Product of the Numbers itself?

The HCF formula is where the product of the LCM and HCF of the two numbers is equivalent to the product of numbers itself. The formula is:

LCM (a & b) × HCF (a & b) = Product of the Numbers ( a & b)

### Find the HCF of 12/5 and 6/15.

According to the HCF formula, HCF = HCF of numerator/LCM of the denominator.

So finding the HCF of the numerators:

12 = 1×2×3×4×6

6 = 1×2×3

Hence HCF of 12 and 6 = 3

Finding the LCM of the denominator

5 = 1×5

15 = 1×3×5

Hence LCM of 5 and 15 = 3×5 = 15

HCF = HCF of numerator/LCM of the denominator

HCF = 3/15 = 1/5

Therefore, the HCF of 12/5 and 6/15 = 1/5

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