HCF and LCM of Prime and Composite Numbers

HCF refers to the Highest Common Factor. It refers to the largest divisor which is common to given set of numbers. LCM refers to the Lowest Common Multiple is the lowest multiple which is common to a given set of numbers.

HCF of a Set of Prime Numbers

The Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), of a set of prime numbers is indeed 1. This property stems from the definition of prime numbers and the fundamental theorem of arithmetic.

Prime numbers are integers greater than 1 that have no positive divisors other than 1 and themselves. Therefore, any two distinct prime numbers have no common factors other than 1 because they cannot be divided evenly by any integer other than 1 and themselves.

When we consider a set of prime numbers, each prime number within the set maintains this property of having no factors other than 1 and itself. As a result, when we look for the highest common factor among them, the only number that evenly divides all of them is 1.

So, the HCF of a set of prime numbers is always 1 due to the unique properties of prime numbers and their inability to be divided by anything other than 1 and themselves.

LCM of a Set of Prime Numbers

The Lowest Common Multiple (LCM) of a set of prime numbers is indeed equal to the product of all distinct numbers from the set. This is a fundamental property that arises from the nature of prime numbers and the definition of LCM.

Since prime numbers are by definition indivisible by any other positive integer except 1 and themselves, when we consider the LCM of a set of prime numbers, each prime number contributes uniquely to the overall product.

When finding the LCM of a set of prime numbers, we essentially look for the smallest multiple that is divisible by all the numbers in the set. Since prime numbers have no common factors except 1, the LCM is simply the product of all the distinct prime numbers in the set.

For example, let’s consider a set of prime numbers: {2, 3, 5}. The LCM of these prime numbers is calculated as:

LCM (2, 3, 5) = 2 * 3 * 5 = 30

In this case, the LCM is indeed equal to the product of the distinct prime numbers in the set, which is 30.

This property holds true for any set of prime numbers, as each prime number contributes uniquely to the LCM without any need for adjustment due to common factors. Therefore, the LCM of a set of prime numbers is equal to the product of all distinct numbers from the set.

HCF & LCM of Composite Numbers

HCF & LCM of Composite Numbers can be find out using Prime Factorization and Division Method.

Prime Factorization of Composite Numbers

Every composite number can be expressed as a product of prime numbers. The process of writing a natural number (other than 1 because it has no prime factors) as a product of prime numbers is known as Prime Factorization and the prime numbers present in it are known as prime factors of the number.

Refer to solved examples 3 and 4 for example.

Note: The prime factorization of a prime number is just the number itself.

Prime and Composite Numbers are commonly used classifications of Natural Numbers based on divisibility and the number of Factors. A Prime Number has only two factors while Composite Numbers have more than two factors. This classification of Numbers makes the study of natural numbers more organized and convenient and is useful in a variety of situations like computer algorithms, biology, understanding of Number Theory, etc.

This article describes what are prime and composite numbers, the types of primes and composite numbers, and tests to check whether a given number is prime or not (primality tests). Finally, a few solved questions, and a few practice problems related to prime and composite numbers are presented.