LCM includes $2^3, 3, 5, 7, 109$ → LCM = $8 \cdot 3 \cdot 5 \cdot 7 \cdot 109 = 72840$

["Understanding LCM: How Multiples and Prime Factors Create the Least Common Multiple", "When solving problems involving fractions, scheduling, or periodic events, the concept of the Least Common Multiple (LCM) is essential. The LCM of two or more numbers is the smallest positive integer that is evenly divisible by all the given numbers. For example, calculating the LCM of (2^3), (3), (5), (7), and (109) reveals a powerful method rooted in prime factorization.", "---", "### What Are the Numbers Involved?", "The numbers in this example are:\n- (2^3 = 8)\n- (3)\n- (5)\n- (7)\n- (109)", "Each of these numbers contributes unique prime factors:", "- (2^3) brings the prime factor 2 with multiplicity 3\n- (3), (5), and (7) are prime numbers themselves\n- (109) is also a prime number", "---", "### Applying Prime Factorization to Compute LCM", "To find the LCM using prime factorization, follow this approach:\nMultiply all distinct prime factors, each raised to their highest exponent found in any of the numbers.", "In this set:\n- The prime factors are (2), (3), (5), (7), and (109)\n- Each appears only once (except 2 raised to the 3rd power, indicating highest multiplicity) \nThus, the LCM is calculated as:", "[\n\ ext{LCM} = 2^3 \ imes 3^1 \ imes 5^1 \ imes 7^1 \ imes 109^1\n]", "Calculating step-by-step:", "[\n2^3 = 8\n]\n[\n8 \ imes 3 = 24\n]\n[\n24 \ imes 5 = 120\n]\n[\n120 \ imes 7 = 840\n]\n[\n840 \ imes 109 = 72840\n]", "---", "### Why (LCM = 8 \cdot 3 \cdot 5 \cdot 7 \cdot 109 = 72840)?", "Using multiplication in this linear sequence efficiently builds the LCM from fundamental building blocks—prime numbers—without needing repeated division or finding individual multiples. This method is especially useful when dealing with more than two numbers or larger exponents.", "---", "### Real-World Applications of LCM", "- Fractions: Finding a common denominator for addition or comparison\n- Repeating Events: Determining when multiple periodic activities (e.g., buses, lights, cycles) align\n- Engineering & Scheduling: Coordinating tasks with different intervals\n- Computer Science: Optimizing memory alignment or algorithmic cycles", "---", "### Summary", "The LCM of (2^3), (3), (5), (7), and (109) is calculated by multiplying these primes with their highest powers:", "[\n\boxed{LCM = 2^3 \cdot 3 \cdot 5 \cdot 7 \cdot 109 = 72840}\n]", "This approach not only gives the correct result efficiently but also deepens your understanding of how prime factorization powers fundamental math concepts. Mastering LCM unlocks improved problem-solving skills across various fields—from basic arithmetic to advanced computational systems.", "---", "Keywords: LCM calculation, least common multiple, prime factorization, math tutorial, LCM examples, LCM of prime powers, LCM step-by-step, LCM 72840, common multiple explained."]