LCM and HCF

The greatest number that divides each of the supplied numbers without leaving any residue is the highest common factor (H.C.F.) of two or more provided numbers. The lowest of the multiple common sets of two or more numbers is called the lowest common multiple (L.C.M.).

In mathematics, L.C.M. stands for Least Common Multiple, while H.C.F. stands for Highest Common Factor. The biggest factor between two or more numbers is defined by the H.C.F., whereas the L.C.M defines the least number that is perfectly divided by two or more numbers. The greatest common factor (GCF) is also known as the highest common factor (H.C.F.), while the least common divisor (L.C.M.) is also known as the least common divisor. 

Difference between L.C.M. & H.C.F.:

H.C.F.: Highest common factor – H.C.F. is the greatest of all the provided numbers’ common factors. The H.C.F. of any given set of data cannot be bigger than the H.C.F. of any of them.

L.C.M.: Least common multiple – L.C.M. is the least of all the provided numbers’ common multiples. The L.C.M. of any given set of data cannot be smaller than any of them.

Methods to calculate LCM & HCF:

Finding the H.C.F. and L.C.M. of numbers can be done in various ways. The following are the most prevalent methods:

• Prime factorization method

• Division method

Calculation of H.C.F. by Prime factorization method: Find the highest common factor (H.C.F.) of 14 and 8 by using the prime factorization method. 

14 = 1 × 2 × 7. 

8 = 1 × 2 × 2 × 2. 

Common factor of 8 and 14 = 1 and 2. 

The product of the lowest powers of factors common to all numbers is H.C.F.

The highest common factor of 8 and 14 = 2. 

Calculation of L.C.M. by Prime Factorization: Find the least common multiple (L.C.M.) of 9 and 15 by using the prime factorization method.

We are identifying and resolving each number’s prime factors.

9 = 3 × 3 = 3².

15 = 3 × 5.

Multiply all the factors with the highest powers.

= 3^2 × 5 = 3 × 3 × 5 = 45.

9 + 15 = 45, which is the required least common multiple.

Calculation of HCF by Division Method:  Find the highest common factor of 18 and 30 by using the division method.

Solution:

Step 1 – Divide 30 by 18.

Step 2 – The first divisor is 18 and the remainder is 12, hence divide 18 by 12.

Step 3 – Divide the second divisor 12 by the second remainder 6.

Step 4 – The remainder becomes 0.

Step 5 – Highest common factor = 6.

Calculation of LCM by Division Method:  Find least common multiple of 20 and 30 by division method.

Solution: Least common multiple (L.C.M) of 20 and 30 = 2 × 2 × 5 × 3 = 60.

The formula for LCM & HCF:

Product of Two numbers = (H.C.F. of any of the two numbers) x (L.C.M. of any of the two numbers)

 Hence, H.C.F. of Two numbers = Product of Two numbers/L.C.M of two numbers

 L.C.M. of two numbers = Product of Two numbers/H.C.F. of Two numbers

The L.C.M. and H.C.F. formula of any two numbers ‘x’ and ‘y’ is given as H.C.F. × L.C.M. = x × y. To put it another way, the H.C.F. and L.C.M. formula asserts that the product of any two numbers is the sum of their H.C.F. and L.C.M. 

Conclusion 

We discussed the Difference between H.C.F. and L.C.M., Methods to calculate H.C.F. and L.C.M., and other related topics through the study material notes on L.C.M. & H.C.F. We also discussed problems to calculate H.C.F. and L.C.M., H.C.F., and lcm questions & solutions to give you proper knowledge.  

The product of the L.C.M. & H.C.F. of any two provided numbers is equal to the product of the supplied numbers, according to the relation between the L.C.M. & H.C.F. of two numbers. Assume that ‘x’ and ‘y’ are the two numbers supplied. L.C.M. (x,y) H.C.F. (x,y) = x× y is the formula that demonstrates the link between their L.C.M. and H.C.F. Let’s use the numbers 10 and 8 as an example. Let’s use the following formula: HCF (10,8) = 10× 8. LCM (10,8) × HCF (10,8) = 10 × 8. The L.C.M. and H.C.F. of 10 and 8 are 40 and 42, respectively. We get 40× 2 = 10× 8 by plugging the numbers into the formula. This demonstrates: 80 = 80.