Solution: To find the greatest common factor (GCF) of 84 and 126, factor both numbers:
["Solution: How to Find the Greatest Common Factor (GCF) of 84 and 126 by Factoring", "When solving mathematics problems involving divisibility or simplifying fractions, one essential concept is the Greatest Common Factor (GCF)—also known as the Greatest Common Divisor (GCD). The GCF of two or more numbers is the largest number that divides each of them without leaving a remainder. In this article, we’ll explore the step-by-step solution for finding the GCF of 84 and 126 by factoring both numbers into their prime components.", "### Step 1: Factor Each Number into Prime Factors", "To find the GCF, begin by factoring both numbers into their prime factors.", "Prime Factorization of 84:\nStart by dividing 84 by the smallest prime, 2:\n- 84 ÷ 2 = 42\n- 42 ÷ 2 = 21\n- 21 ÷ 3 = 7\n- 7 is a prime number.", "So,\n[ 84 = 2 \ imes 2 \ imes 3 \ imes 7 = 2^2 \ imes 3 \ imes 7 ]", "Prime Factorization of 126:\nDivide 126 by 2:\n- 126 ÷ 2 = 63\n- 63 ÷ 3 = 21\n- 21 ÷ 3 = 7\n- 7 is prime.", "Thus,\n[ 126 = 2 \ imes 3 \ imes 3 \ imes 7 = 2 \ imes 3^2 \ imes 7 ]", "### Step 2: Identify Common Prime Factors", "Now, compare the prime factorizations:\n- 84: ( 2^2 \ imes 3 \ imes 7 )\n- 126: ( 2 \ imes 3^2 \ imes 7 )", "The common prime factors are:\n- ( 2^1 ) (minimum exponent between 84 and 126 is 1)\n- ( 3^1 )\n- ( 7^1 )", "### Step 3: Multiply the Common Prime Factors", "Multiply the lowest-powered common primes:\n[\n\ ext{GCF} = 2^1 \ imes 3^1 \ imes 7^1 = 2 \ imes 3 \ imes 7 = 42\n]", "### Final Result", "The greatest common factor of 84 and 126 is 42.", "---", "Factoring numbers is a reliable method for finding the GCF, especially useful for understanding divisibility, simplifying fractions, and solving math problems efficiently. This process encourages a deeper grasp of number relationships and lays a solid foundation for working with larger or more complex numbers in algebra and number theory.", "---", "Tip: Practice factoring different numbers and use the prime factor listing method to confirm results. Always take the lowest power of shared primes to determine the GCF efficiently.", "---", "Understanding the GCF through prime factorization not only solves current problems but also strengthens foundational math skills for future learning."]
