Solution: Factorize $ 210 = 2 \cdot 3 \cdot 5 \cdot 7 $ and $ 294 = 2 \cdot 3 \cdot 7^2 $. The GCD is the product of the smallest powers of common primes: $ 2^1 \cdot 3^1 \cdot 7^1 = 42 $. oxed{42}
["Factorization of Key Composite Numbers and the Greatest Common Divisor (GCD) Explained", "Understanding how to factorize composite numbers is a fundamental skill in number theory and essential for solving problems such as finding the Greatest Common Divisor (GCD). In this article, we examine two important factorizations— ( 210 = 2 \cdot 3 \cdot 5 \cdot 7 ) and ( 294 = 2 \cdot 3 \cdot 7^2 )—and demonstrate how to compute their GCD using prime factorization.", "---", "### Factorization of 210 and 294", "Begin with the prime factorization of 210:", "[\n210 = 2 \cdot 3 \cdot 5 \cdot 7\n]", "This expression breaks 210 down entirely into its prime components, each appearing only once—no repeated factors.", "Now, factorize 294:", "[\n294 = 2 \cdot 3 \cdot 7^2\n]", "Here, 294 factors into 2, 3, and (7^2)—noting that 7 appears squared, reflecting the higher power in its prime decomposition.", "---", "### Determining the Greatest Common Divisor (GCD)", "The GCD of two integers is found by taking the product of each common prime factor raised to the smallest power present in both factorizations.", "Both integers share the primes: 2, 3, and 7.", "Compare the exponents:", "- Power of 2: ( \min(1, 1) = 1 ) → ( 2^1 )\n- Power of 3: ( \min(1, 1) = 1 ) → ( 3^1 )\n- Power of 7: ( \min(1, 2) = 1 ) → ( 7^1 )", "Multiply these together:", "[\n\ ext{GCD} = 2^1 \cdot 3^1 \cdot 7^1 = 2 \cdot 3 \cdot 7 = 42\n]", "Thus, the greatest common divisor of 210 and 294 is:", "[\n\boxed{42}\n]", "---", "### Why This Method Matters", "Factorizing numbers into primes not only reveals foundational building blocks of integers but also enables efficient computation of GCD, LCM, and other number-theoretic operations. This approach simplifies complex problems in cryptography, algebra, and programming.", "Understanding these concepts is especially useful when analyzing number properties or solving equations involving divisors and multiples.", "---", "### Summary", "- Prime factorization of 210: ( 2 \cdot 3 \cdot 5 \cdot 7 )\n- Prime factorization of 294: ( 2 \cdot 3 \cdot 7^2 )\n- The GCD is computed by taking the minimum exponent for each common prime:\n ( 2^1 \cdot 3^1 \cdot 7^1 = 42 )", "So, the solution to factorizing these numbers and finding their GCD is clearly:", "[\n\boxed{42}\n]"]
