Solution: To find the GCF of 42 and 63, factor both numbers. $42 = 2 \times 3 \times 7$ and $63 = 3^2 \times 7$. The common prime factors are $3$ and $7$. Multiply them: $3 \times 7 = 21$. $\boxed{21}$

["Finding the GCF of 42 and 63: A Step-by-Step Solution Using Factorization", "Understanding how to find the Greatest Common Factor (GCF) is essential for mastering number theory and solving problems in algebra, math education, and real-world applications. One of the most effective methods to determine the GCF of two numbers is factoring both numbers into their prime components. In this article, we’ll explore how to find the GCF of 42 and 63 using this powerful factorization technique.", "### What is GCF?", "The GCF, also known as the Greatest Common Divisor, is the largest positive integer that divides two or more numbers without leaving a remainder. It plays a crucial role in simplifying fractions, solving equations, and reducing algebraic expressions.", "### Step 1: Prime Factorization of 42 and 63", "To find the GCF, we begin by breaking each number into its prime factors.", "- Factor 42:\n 42 is an even number, so it’s divisible by 2:\n $ 42 = 2 \ imes 21 $\n Now factor 21:\n $ 21 = 3 \ imes 7 $\n So, the complete prime factorization is:\n $$ 42 = 2 \ imes 3 \ imes 7 $$", "- Factor 63:\n 63 is not divisible by 2, but it is divisible by 3:\n $ 63 = 3 \ imes 21 $\n Continue factoring:\n $ 21 = 3 \ imes 7 $\n Thus,\n $$ 63 = 3^2 \ imes 7 $$", "### Step 2: Identify Common Prime Factors", "Once both numbers are expressed in prime factors, identify the primes that both numbers share:", "- $ 42 = 2 \ imes 3 \ imes 7 $\n- $ 63 = 3^2 \ imes 7 $", "The common prime factors are 3 and 7.", "### Step 3: Multiply Common Prime Factors", "Now multiply the shared primes, using the lowest exponent for any common prime (though here the exponents are straightforward):", "$$\n\ ext{GCF} = 3 \ imes 7 = 21\n$$", "### Conclusion", "The GCF of 42 and 63 is $ \boxed{21} $. This method of factoring not only gives us an accurate result quickly but also deepens our understanding of number relationships. Whether in elementary math, SAT preparation, or advanced math, mastering GCF through factorization builds a solid foundation for further numerical reasoning.", "---", "Key Takeaways:\n- Break numbers into prime factors for easy GCF calculation\n- Look for common primes and multiply them\n- This technique is accurate, efficient, and widely applicable", "Happy calculating! Using prime factorization to find GCF simplifies complex problems—start practicing today!"]