Solution: Factorize 98 as $2 imes 7^2$ and 126 as $2 imes 3^2 imes 7$. The GCD is the product of the lowest powers: $2 imes 7 = 14$. Thus, the GCD is $oxed{14}$.
["Factorization Insights: Unlocking GCD with Prime Decomposition", "Understanding the Greatest Common Divisor (GCD) is fundamental in number theory, offering powerful tools for simplifying fractions, solving equations, and optimizing algorithms. One effective method to compute the GCD involves breaking numbers into their prime factors—a process that reveals the core building blocks of integers.", "In this article, we explore the factorization of two key numbers—98 and 126—and demonstrate how prime decomposition leads directly to an accurate GCD calculation.", "---", "### Factoring 98: $ 2 \ imes 7^2 $", "To factor 98, we start with prime number testing:", "- 98 is even, so divisible by 2:\n $ 98 = 2 \ imes 49 $", "- Next, factor 49:\n $ 49 = 7 \ imes 7 = 7^2 $", "Thus, the complete prime factorization of 98 is:\n$$\n98 = 2 \ imes 7^2\n$$", "---", "### Factoring 126: $ 2 \ imes 3^2 \ imes 7 $", "Now examine 126:", "- 126 is even:\n $ 126 = 2 \ imes 63 $", "- Factor 63: divisible by 3:\n $ 63 = 3 \ imes 21 $", "- Factor 21:\n $ 21 = 3 \ imes 7 $", "Putting it all together:\n$$\n126 = 2 \ imes 3^2 \ imes 7\n$$", "---", "### Computing the GCD Using Lowest Powers", "The GCD of two numbers is determined by taking each prime present in both factorizations and using the lowest exponent of that prime.", "- The prime 2 appears in both with exponent 1:\n $ \min(1, 1) = 1 $", "- The prime 3 is absent in 98, so excluded", "- The prime 7 appears in both as $ 7^1 $ and $ 7^2 $; use $ 7^1 $", "Thus:\n$$\n\ ext{GCD} = 2^1 \ imes 7^1 = 2 \ imes 7 = 14\n$$", "---", "### Final Result: $ \boxed{14} $", "By decomposing 98 and 126 into prime factors and identifying the shared lowest powers, we confidently determine that their greatest common divisor is $ \boxed{14} $. This method not only simplifies GCD computation but also strengthens foundational knowledge in factorization and number theory.", "---", "Keywords: GCD calculation, prime factorization, factor 98, factor 126, lowest common multiple insight, number theory fundamentals, mathematical decomposition, algorithm explanation, interactive learning math.", "Understanding and practicing factorization equips learners and professionals alike with a reliable strategy for simplifying complex numerical relationships across mathematics and computer science."]
