Properties of HCF And LCM

HCF and LCM are considered as one of the most important topics in mathematics as they are very useful in solving problems related to time and work, time and distance, pipes and cisterns, etc. Knowing about L.C.M and H.C.F of two or more numbers helps in finding out quick solutions and thus reduces the time spent on calculation. HCF can be useful in solving many mathematical problems like calculating the largest tape to measure the land, largest size of tile, etc. LCM can be useful in solving many mathematical problems related to racetracks and traffic lights. LCM is also useful in computer science, for designing encoded messages using cryptography.

Before understanding the Properties of LCM And HCF, let us understand the basic concepts & definitions of HCF And LCM.

Definition of LCM: LCM stands for Lowest or Least Common Multiple. The LCM of two or more numbers can be defined as the smallest positive integer that is divisible by all the given numbers. For instance, find the LCM of 16 and 20. The LCM of 16 and 20 can be calculated as 2 × 2 × 2 × 2 × 5 = 80. Here, 80 is the LCM of numbers 16 and 20.

Property 1: The HCF of any given numbers is never greater than any of the numbers

The HCF of any given number is always less than or equal to any of the given numbers. For instance, Consider 6 and 27. Their HCF is 3, which is a number that is lesser than both the numbers 6 and 27. Consider another example, say we have two numbers 4 and 20. Here the HCF is 4, which is equal to one of the numbers

Example: The HCF of 16, 18, and 24 is found to be 2. Here, 2 is less than all the given numbers

Property 2: The HCF of co-prime numbers is always 1

When both the numbers are co-prime, they have no common factor other than 1. Hence, 1 is the only common factor between the two given numbers. So, the HCF of two co-prime numbers is always 1. Also, any two successive numbers/ integers are always co-prime. For instance, take any consecutive numbers such as 3, 4, or 2, 3, or 5, 6, and so on; they have 1 as their HCF. For instance, consider two co-prime numbers, 3 and 7. The HCF of 3 and 7 is 1.

Property 3: The product of LCM and HCF of any two given natural numbers is always equal to the product of those given numbers.

LCM x HCF = Product of the numbers. 

Suppose A and B are two numbers, then LCM of (A and B) × HCF of (A and B) = Product of (A and B) or, HCF of (A and B) × LCM of (A and B) = Product of (A and B) or, LCM of (A and B) = Product of (A and B) / HCF of (A and B) or, HCF of (A and B) = Product of (A and B) / LCM of (A and B)

Note: This property is applicable for only two numbers.

For instance, If 3 and 8 are two numbers. LCM of 3 and 8 = 24 and HCF of 3 and 8 = 1. LCM (3,8) × HCF (3,8) = 24 × 1 = product of the numbers (3 × 8) = 24

Property 4: LCM of given co-prime numbers is always equal to the product of the numbers

Co-primes are the pair of numbers whose common factor includes just 1. Therefore, the LCM of two co-primes is always the product of these co-prime numbers. The LCM of a and b where a and b are co-primes is a × b.

For instance, Consider two co-prime numbers 3 and 5. The LCM of 3 and 5 is 3 × 5 × 1 = 15, which is the product of the co primes.

Property 5: H.C.F. and L.C.M. of Fractions

Let’s consider two fractions (p/q) and (r/s). To find LCM and HCF of (p/q) and (r/s) the generalized formula will be:

H.C.F = HCF of numerators / LCM of denominators

L.C.M = LCM of numerators / HCF of denominators

Example: For instance, Find the HCF and LCM of 4/9 and 6/21.

Property 6: The LCM of given numbers is not less than any of the given numbers.

The LCM is the smallest number that both the given numbers divide into. However, it will be greater than at least one (or often both) of the given numbers. Additionally, if a number is the factor of another number, then their LCM is the greater number itself. For instance, LCM of 8 and 12.

L.C.M. of 8 and 12 is 24 which is not less than any of the given numbers. For instance, the LCM of 8 and 16 is 16, the greater number itself.

Properties of HCF And LCM Related Articles

Check out these interesting articles to learn more about the properties of HCF and LCM and their related topics

• Multiples
• Factors
• Highest Common Factor (HCF)
• Least Common Multiple (LCM)
• Co-prime numbers

Important Notes

• H.C.F of two or more numbers is considered to be the greatest among all the common factors of them
• The LCM of two or more numbers can be defined as the smallest positive integer that is divisible by all the given numbers
• The HCF of co-prime numbers is always 1
• LCM of given co-prime numbers is always equal to the product of the numbers

What are the Various Methods to Find the Least Common Multiple of Two or More Numbers?

The various methods to find the least common multiple are: using prime factorization, using repeated division, and using the listing multiples method. We can use any one of the methods to find LCM based on the complexity of the numbers.

Which of the Properties of HCF and LCM Defines the Formula to Calculate the HCF and LCM of Fractions?

Let’s consider two fractions (p/q) and (r/s). To find LCM and HCF of (p/q) and (r/s), the generalized formula will be HCF of Fractions is given as HCF of numerators / LCM of denominators. LCM of Fractions is given as LCM of numerators / HCF of denominators. These two formulae can be used to find the HCF and LCM of fractions.

Which of the Properties of HCF and LCM is Applicable Only for Two Numbers?

The product of LCM and HCF of any two given natural numbers is said to be equal to the product of the given numbers. This is the property that is applicable only for two numbers.

Which Property of HCF and LCM Talk About Prime Numbers?

Co-primes are the pair of numbers whose common factor includes just 1. Therefore, the LCM of two co-primes is always the product of these co-prime numbers. The LCM of a and b where a and b are co-primes is a x b.

What are the Properties of LCM?

LCM of given co-prime numbers is always equal to the product of the numbers. And, the LCM of given numbers is not less than any of the given numbers.

What is the Use of HCF and LCM?

HCF is used whenever the situation talks about the maximum or greatest. Also, only HCF is used whenever the question is related to distribution into groups or classification. LCM is used whenever the situation talks about the minimum or smallest. Also, only LCM is used whenever the word ‘simultaneous’ or ‘together’ is used.