Question: What is the least common multiple of $ 2^4 \cdot 3 $ and $ 2^2 \cdot 3^3 \cdot 5 $?
["Understanding the Least Common Multiple (LCM) of $ 2^4 \cdot 3 $ and $ 2^2 \cdot 3^3 \cdot 5 $: A Clear & Step-by-Step Guide", "When dealing with fractions, rational numbers, or repeated prime factors, one of the most essential concepts is the Least Common Multiple (LCM). But what happens when the numbers are expressed in their prime factorization form, such as $ 2^4 \cdot 3 $ and $ 2^2 \cdot 3^3 \cdot 5 $? How do we calculate their LCM efficiently? This article explains everything you need to know about finding the LCM of these expressions clearly and accurately.", "---", "### What Is the Least Common Multiple (LCM)?", "The LCM of two or more integers (or algebraic expressions) is the smallest number (or expression) that both or all of them divide evenly without leaving a remainder. In prime factorized form, the LCM takes the highest power of each prime factor found in the inputs.", "---", "### The Numbers in Factored Form", "We are given two values:", "- $ a = 2^4 \cdot 3 $\n- $ b = 2^2 \cdot 3^3 \cdot 5 $", "Both are already expressed using prime factorizations:\n- $ 2^4 \cdot 3 = 2^4 \ imes 3^1 $\n- $ 2^2 \cdot 3^3 \cdot 5 = 2^2 \ imes 3^3 \ imes 5^1 $", "---", "### Step-by-Step Guide to Calculating the LCM", "#### Step 1: Identify all prime factors involved\nFrom both expressions, the prime factors present are:\n- $ 2 $,\n- $ 3 $,\n- $ 5 $.", "#### Step 2: Determine the highest power of each prime\nFor each prime, take the maximum exponent appearing in either factorization:", "- For $ 2 $: max(4, 2) = 4 → use $ 2^4 $\n- For $ 3 $: max(1, 3) = 3 → use $ 3^3 $\n- For $ 5 $: max(0, 1) = 1 → use $ 5^1 $", "#### Step 3: Multiply these highest powers together\n[\n\ ext{LCM} = 2^4 \cdot 3^3 \cdot 5\n]", "This is the smallest number divisible by both original values.", "---", "### Final Answer", "The least common multiple of $ 2^4 \cdot 3 $ and $ 2^2 \cdot 3^3 \cdot 5 $ is:", "[\n\boxed{2^4 \cdot 3^3 \cdot 5}\n]", "Numerically, this equals $ 16 \cdot 27 \cdot 5 = 2160 $, but the factored form is often preferred in mathematical applications.", "---", "### Why This Matters", "Understanding how to compute LCMs using prime factorizations strengthens your ability to work with fractions, ratios, and algebraic expressions. Whether you’re simplifying fractions, solving real-world problems involving cycles or schedules, or exploring divisibility rules, mastering LCM ensures clearer and more efficient problem-solving.", "---", "Smart Tip: When dealing with prime factorized numbers, always\n1. List all unique primes,\n2. Take the highest power of each,\n3. Multiply them — and you’ve found the LCM!", "---", "### Further Reading\n- How to Find GCD Using Prime Factorization\n- Practical Applications of LCM in Math and Engineering\n- Prime Factorization Techniques for Complex Expressions", "---", "This SEO article is optimized with key phrases like “LCM of prime factorized expressions,” “how to calculate LCM with exponents,” and “prime factorization method,” making it highly visible for learners searching for clear math guidance on least common multiple."]
