Solution: We use the prime factorization method. \(68 = 2^2 \times 17\) and \(102 = 2 \times 3 \times 17\). The common prime factors are 2 and 17. Taking the lowest powers, we get \(2^1 \times 17^1 = 34\). Thus, the greatest common factor is \(\boxed{34}\).

["Finding the Greatest Common Factor with Prime Factorization: A Step-by-Step Solution", "Understanding the greatest common factor (GCF) is essential in algebra and number theory. One of the most reliable methods to compute GCF is prime factorization. This article explains, with a concrete example, how to find the GCF using prime factorization—perfect for students and math enthusiasts.", "---", "### What is the Greatest Common Factor (GCF)?", "The GCF of two or more integers is the largest number that divides each of them without leaving a remainder. It’s also known as the greatest common divisor (GCD). Whether solving equations, simplifying fractions, or analyzing patterns, knowing the GCF helps simplify complex problems efficiently.", "---", "### Why Use Prime Factorization?", "Prime factorization breaks down each number into a product of prime numbers. Once numbers are expressed as powers of primes, identifying common factors becomes straightforward—especially when selecting the lowest powers of shared primes. This method guarantees accuracy and strengthens foundational number sense.", "---", "### Step-by-Step Example: Find the GCF of 68 and 102", "Let’s apply prime factorization to determine the GCF of 68 and 102—two seemingly unrelated numbers.", "#### Step 1: Prime Factorization of 68\nWe begin by factoring 68 into its prime components:\n[ 68 \div 2 = 34 ]\n[ 34 \div 2 = 17 ]\n17 is a prime number, so:\n[ 68 = 2^2 \ imes 17^1 ]", "#### Step 2: Prime Factorization of 102\nNext, factor 102 using prime division:\n[ 102 \div 2 = 51 ]\n51 is not divisible by 2 but divisible by 3:\n[ 51 \div 3 = 17 ]\n17 remains prime:\n[ 102 = 2^1 \ imes 3^1 \ imes 17^1 ]", "#### Step 3: Identify Common Prime Factors\nNow compare the factorizations:\n- Factor 68: ( 2^2 \ imes 17^1 )\n- Factor 102: ( 2^1 \ imes 3^1 \ imes 17^1 )", "The common prime factors are 2 and 17. To compute the GCF, take the lowest exponent of each common prime:\n- For 2: minimum power is ( 2^1 )\n- For 17: minimum power is ( 17^1 )", "#### Step 4: Multiply the Lowest-Powered Common Primes\n[ \ ext{GCF} = 2^1 \ imes 17^1 = 2 \ imes 17 = 34 ]", "---", "### Conclusion: The Greatest Common Factor Is 34", "Using the prime factorization method, we confidently determined that:\n[ \boxed{34} ]\nis the GCF of 68 and 102.", "This systematic approach—break down numbers, identify shared primes, use lowest powers—is powerful and scalable to larger numbers. Mastering prime factorization not only simplifies GCF calculations but also deepens your grasp of number theory.", "---", "Try It Now: Practice with other pairs like 96 and 72 or 45 and 75 using the same method. With consistent practice, prime factorization becomes a quick and reliable tool in your math toolkit!"]