Solution: Prime factorization: $108 = 2^2 \cdot 3^3$, $144 = 2^4 \cdot 3^2$. GCD takes the lowest powers: $2^2 \cdot 3^2 = 4 \cdot 9 = 36$.
["Understanding Prime Factorization and How GCD Works: A Clear Guide with Real Examples", "When learning about numbers and their properties, prime factorization and the Greatest Common Divisor (GCD) are essential concepts that simplify complex problems. One powerful approach to finding the GCD of two numbers is using their prime factorizations. This method becomes particularly clear when we examine examples like $108 = 2^2 \cdot 3^3$ and $144 = 2^4 \cdot 3^2$. By exploring how prime factorization and GCD calculations work step-by-step, anyone can master these foundational math tools.", "---", "### What Is Prime Factorization?", "Prime factorization is the process of breaking down a composite number into a product of prime numbers. Every integer greater than 1 can be uniquely expressed as a product of primes raised to some powers. This uniqueness is known as the Fundamental Theorem of Arithmetic.", "Example:\n$$\n108 = 2 \cdot 54 = 2 \cdot 2 \cdot 27 = 2^2 \cdot 3^3\n$$\n$$\n144 = 2 \cdot 72 = 2 \cdot 2 \cdot 36 = 2 \cdot 2 \cdot 2 \cdot 18 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 9 = 2^4 \cdot 3^2\n$$", "These representations break each number into its prime building blocks:\n- $108 = 2^2 \cdot 3^3$\n- $144 = 2^4 \cdot 3^2$", "---", "### How GCD Uses Prime Factorization", "The Greatest Common Divisor (GCD) is the largest number that divides two or more integers without leaving a remainder. Using prime factorization, finding the GCD is straightforward:", "1. Factor both numbers into primes (as shown above).\n2. Identify common prime factors.\n3. Take the lowest power of each common prime.\n4. Multiply these together to get the GCD.", "---", "### Let’s Apply This to 108 and 144", "Given:\n- $108 = 2^2 \cdot 3^3$\n- $144 = 2^4 \cdot 3^2$", "Step 1: List the prime factors common to both numbers. Here, the common primes are $2$ and $3$.", "Step 2: For $2$, the powers are $2^2$ (in 108) and $2^4$ (in 144). The lowest power is $2^2$.\nFor $3$, the powers are $3^3$ (in 108) and $3^2$ (in 144). The lowest power is $3^2$.", "Step 3: Multiply these together:\n$$\n\ ext{GCD} = 2^2 \cdot 3^2 = 4 \cdot 9 = 36\n$$", "---", "### Why This Method Works", "By taking the lowest exponent for each shared prime factor, we ensure the result divides both numbers evenly. This technique eliminates guesswork and reduces complex divisibility rules to simple arithmetic—ideal for students and problem solvers alike.", "---", "### Final Thoughts", "Understanding prime factorization and applying it to compute the GCD transforms abstract math into a clear, logical process. Whether tackling homework, exams, or real-world problems involving ratios, fractions, or optimization, this method strengthens number sense and computational fluency.", "Next time you see numbers like $108$ and $144$, break them down, compare their prime exposures, and apply the lowest power rule to find their GCD—36—confidently and correctly.", "---", "Keywords for SEO:\nprime factorization, GCD, greatest common divisor, mathematical methods, prime factor example, solution prime factorization, 108 factorization, 144 factorization, math explanation, divisibility rules, exponents in math, prime factor decomposition", "Meta description:\nLearn how prime factorization simplifies GCD calculation with clear examples. See why $108 = 2^2 \cdot 3^3$ and $144 = 2^4 \cdot 3^2$ leads to $ \ ext{GCD} = 2^2 \cdot 3^2 = 36 $. Step-by-step guide for students and math enthusiasts."]
