Whole Numbers, HCF, and LCM Concepts

Whole Numbers

• Whole numbers are the set of numbers starting from zero and continuing infinitely in the positive direction.
• They include zero and all positive integers (1, 2, 3,...).
• Whole numbers do not include negative numbers or fractions.
• They are used for counting and measuring.

Highest Common Factor (HCF)

• The HCF of two or more numbers is the largest positive integer that divides each of the given numbers without leaving a remainder.
• It is also known as the Greatest Common Divisor (GCD).
• Finding the HCF involves finding the prime factors of each number and then identifying the common prime factors with the lowest powers.
• Example: The HCF of 12 and 18 is 6 because 6 is the largest number that divides both 12 and 18 evenly.
• Methods to find HCF include prime factorization, listing factors, and the Euclidean algorithm.

Least Common Multiple (LCM)

• The LCM of two or more numbers is the smallest positive integer that is a multiple of each of the given numbers.
• It is often found by identifying the prime factorization of each number, then choosing the highest powers of all the prime factors that appear in the factorization, and multiplying these together.
• Example: The LCM of 4 and 6 is 12 because 12 is the smallest number that is a multiple of both 4 and 6.
• Methods to find the LCM include prime factorization, listing multiples, and using the relationship between the HCF and LCM.

Prime Numbers

• A prime number is a whole number greater than 1 that has only two factors: 1 and itself.
• The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, and so on.
• 1 is not a prime number because it only has one factor (itself).
• 2 is the only even prime number because all other even numbers are divisible by 2.
• A composite number has more than two factors.
• A prime number can be expressed as a product of only 1 and itself.
• Prime numbers play a crucial role in number theory and cryptography.
• Prime factorization is the process of writing a composite number as a product of its prime factors.
• Example of a composite number is 15 (15 = 3 x 5)
• The Sieve of Eratosthenes is a method for identifying prime numbers up to a given limit.
• Prime numbers are used in many applications, including cryptography, number theory, and computer science.