Welcome to our lesson on HCF and LCM. HCF, or Highest Common Factor, is the largest number that can divide two or more numbers without leaving a remainder. LCM, or Lowest Common Multiple, is the smallest number that is a multiple of two or more given numbers. Let's explore these concepts using the example of numbers twelve and eighteen.
Now let's learn the prime factorization method to find HCF and LCM. First, we find the prime factorization of each number. Twelve equals two squared times three to the power one. Eighteen equals two to the power one times three squared. For HCF, we take the lowest powers of common prime factors. For LCM, we take the highest powers of all prime factors involved.
Let's work through a complete example with numbers twenty-four and thirty-six. First, we find the prime factorization. Twenty-four equals two cubed times three. Thirty-six equals two squared times three squared. The common prime factors are two and three. For HCF, we take the lowest powers: two squared times three equals twelve. For LCM, we take the highest powers: two cubed times three squared equals seventy-two.
Here's an important relationship between HCF and LCM. For any two numbers a and b, the product of their HCF and LCM equals the product of the original numbers. Let's verify this with our example of twelve and eighteen. HCF is six, LCM is thirty-six. Six times thirty-six equals two hundred sixteen. Twelve times eighteen also equals two hundred sixteen. The formula is verified!
To summarize what we have learned about HCF and LCM: HCF is the largest number that divides all given numbers. LCM is the smallest number that is a multiple of all given numbers. Prime factorization provides a systematic method for calculation. For HCF, use the lowest powers of common factors. For LCM, use the highest powers of all factors. Always remember the verification formula.