3.2 Use prime factors to determine the HCF of \( 28 ; 196 \) and 280

To find the highest common factor (HCF) using prime factors, let's first break down each number into its prime components: - \( 28 \): \( 28 = 2^2 \times 7^1 \) - \( 196 \): \( 196 = 2^2 \times 7^2 \) - \( 280 \): \( 280 = 2^3 \times 5^1 \times 7^1 \) Now, identify the common prime factors: \( 2^2 \) and \( 7^1 \). The HCF is found by taking the lowest power of each common factor. So, \[ HCF = 2^2 \times 7^1 = 4 \times 7 = 28 \] Thus, the HCF of \( 28, 196, \) and \( 280 \) is \( 28. \) Understanding the use of prime factorization isn’t just about math class; it's a powerful tool in various real-life situations! For example, if you're planning a party and need to divide drinks or treats into the same-sized portions, knowing the HCF can help you determine how to distribute items evenly among guests without leftovers. Gotta keep those snacks fair and square! If you’re curious about the role of HCF in other fields, consider diving into topics like cryptography or computer algorithms! HCF and its concepts are foundational in encryption methods and optimization algorithms. You can explore titles like “The Art of Computer Programming” by Donald Knuth or “Elementary Number Theory” by David M. Burton for deeper insights into how these mathematical principles are applied in technology and computing. It’s a whole world of numbers waiting to be discovered!