Let GCF(a, b) be the abbreviation for the greatest common factor of a and b, and let LCM(c, d) be the abbreviation for the least common multiple of c and d. What is GCF(LCM(8, 14), LCM(7, 12))?
I need to find the greatest common factor of two least common multiples, so I will start by finding the least common multiples. To find the least common multiple of two numbers, I can use prime factorization and take the highest power of each prime factor that appears in either number. For example, 8 = 2^3 and 14 = 2 * 7, so LCM(8, 14) = 2^3 * 7 = 56. Similarly, 7 = 7 and 12 = 2^2 * 3, so LCM(7, 12) = 2^2 * 3 * 7 = 84. Now I have GCF(LCM(8, 14), LCM(7, 12)) = GCF(56, 84). To find the greatest common factor of two numbers, I can also use prime factorization and take the lowest power of each prime factor that appears in both numbers. For example, 56 = 2^3 * 7 and 84 = 2^2 * 3 * 7, so GCF(56, 84) = 2^2 * 7 = 28. Therefore, the final answer is 28. 