termine the highest common factor (HCF) of 6480 and 1728

To find the highest common factor (HCF) of 6480 and 1728, we can use the prime factorization method. The prime factorization of 6480 is \(2^4 \times 3^4 \times 5\) and for 1728, it is \(2^6 \times 3^3\). The HCF is obtained by taking the lowest powers of all prime factors present in both numbers, which gives us \(2^4 \times 3^3 = 432\). So, the HCF of 6480 and 1728 is 432! Now, did you know that this method of using prime factorization is not just for finding HCF? It's also handy for simplifying fractions! By breaking down the numerator and denominator into their prime components, you can find the HCF quickly, which will help you reduce the fraction to its simplest form. Additionally, the concept of HCF isn’t just a classroom staple; it has real-world implications! It’s especially relevant in situations like evenly distributing items. For example, if you have 6480 apples and 1728 oranges, finding the HCF tells you the largest number of identical boxes you can create without leftovers. Isn’t that a juicy way to use math?