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: 

1. LCM × HCF = Product of the Numbers. Suppose P and Q are two numbers, then - LCM (P & Q) × HCF (P & Q) = P × Q. 
2. HCF of co-prime numbers is always 1. Therefore, LCM of Co-prime Numbers = Product of the numbers
3. 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.
4. To find the HCF of fractions, we use HCF of fractions = HCF of numerators/LCM of the denominators 
5. To find the greatest number that will exactly divide p, q, and r. Required number = HCF of p, q, and r
6. 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)
7. 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