Solution: We find $ \gcd(72, 108, 180) $.

["Title: How to Find $ \gcd(72, 108, 180) $: Step-by-Step Solution", "When tasked with finding the greatest common divisor (GCD) of three numbers — specifically $ \gcd(72, 108, 180) $ — many learners feel overwhelmed. But with a clear, systematic approach, solving this problem becomes simple and intuitive. In this SEO-optimized article, we’ll break down the solution and explain how to efficiently compute the GCD of multiple numbers.", "---", "### What Is GCD and Why Does It Matter?", "The greatest common divisor (also called GCD or HD, short for highest divisor) of a set of integers is the largest positive integer that divides each number without leaving a remainder. Computational geometry, cryptography, and algorithm design frequently use GCD calculations, making mastering this concept essential.", "---", "### Step-by-Step Solution: Finding $ \gcd(72, 108, 180) $", "The GCD of more than two numbers can be computed using the associative property:", "$$\n\gcd(a, b, c) = \gcd(\gcd(a, b), c)\n$$", "Step 1: Compute $ \gcd(72, 108) $", "We use the Euclidean algorithm to find the GCD of 72 and 108.", "- $ 108 = 72 \ imes 1 + 36 $\n- $ 72 = 36 \ imes 2 + 0 $", "Since the remainder is 0, the GCD is the last non-zero remainder:\n$$\n\gcd(72, 108) = 36\n$$", "Step 2: Compute $ \gcd(36, 180) $", "Now apply the Euclidean algorithm again with 36 and 180:", "- $ 180 = 36 \ imes 5 + 0 $", "Since the remainder is 0 immediately,\n$$\n\gcd(36, 180) = 36\n$$", "---", "### Final Result", "Putting it together:", "$$\n\gcd(72, 108, 180) = \gcd(\gcd(72, 108), 180) = \gcd(36, 180) = 36\n$$", "✅ So, $ \gcd(72, 108, 180) = 36 $", "---", "### Bonus Tips for Faster GCD of Multiple Numbers", "1. Factorization Method\n Factor each number:\n - $ 72 = 2^3 \cdot 3^2 $\n - $ 108 = 2^2 \cdot 3^3 $\n - $ 180 = 2^2 \cdot 3^2 \cdot 5 $\n Take the minimum exponent for each common prime:\n - $ 2^2 $, $ 3^2 $, no 5 common — so $ GCD = 2^2 \cdot 3^2 = 4 \cdot 9 = 36 $", "2. Use Prime Factorization or Euclidean Algorithm Based on Numbers’ Size\n For smaller numbers, factoring is efficient; for larger ones or more numbers, the Euclidean method saves time.", "---", "### Why This Matters in Real Applications", "Computing $ \gcd(72, 108, 180) = 36 $ is not just an academic exercise. It helps in:", "- Simplifying fractions involving three denominators\n- Reducing data transmission sizes in digital signal processing\n- Solving modular arithmetic problems in cryptography", "---", "### Summary", "- Fundamental Theorem: $ \gcd(a, b, c) = \gcd(\gcd(a, b), c) $\n- Step-by-step method ensures accuracy\n- Use prime factorization or Euclidean algorithm depending on context\n- $ \gcd(72, 108, 180) = 36 $", "### Optimize Your Understanding", "For quick review and practice, try computing GCDs of other triplets like $ \gcd(24, 36, 60) $ or $ \gcd(48, 72, 108) $. Mastering these steps builds a strong foundation in number theory and enhances your problem-solving versatility.", "---", "Keywords: GCD of multiple numbers, how to find gcd(72, 108, 180), Euclidean algorithm, gcd method, gcd calculation, number theory, simplifying GCD, math tutorial, integer divisors, math problem solver.", "---", "By following this step-by-step solution and understanding the underlying principles—like the associative property and prime factorization —you’ll confidently solve $ \gcd(72, 108, 180) $ and other GCD challenges. Always verify using multiple methods to reinforce your learning!"]