Solution: Factorize each number: $ 12 = 2^2 \times 3 $, $ 18 = 2 \times 3^2 $, and $ 30 = 2 \times 3 \times 5 $. The LCM is the product of the highest powers of all primes: $ 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180 $.

February 22, 2026

$Solution: Factorize each number: $ 12 = 2^2 \times 3 $, $ 18 = 2 \times 3^2 $, and $ 30 = 2 \times 3 \times 5 $. The LCM is the product of the highest powers of all primes: $ 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180 $.$

["Understanding Factorization and Finding the Least Common Multiple (LCM)", "Factorizing numbers into their prime components is a fundamental skill in mathematics, especially in number theory and algebra. It not only simplifies expressions but also helps in solving problems involving multiples and divisors. In this article, we’ll explore how to factorize key numbers and use their prime factorizations to find the Least Common Multiple (LCM).", "### Prime Factorization of Key Numbers", "Let’s begin by factorizing the numbers 12, 18, and 30:", "- 12 can be expressed as:\n $ 12 = 2^2 \ imes 3 $\n This means 12 is made by multiplying 2 by itself twice (or $ 2 \ imes 2 $) and then by 3.", "- 18 factors into:\n $ 18 = 2 \ imes 3^2 $\n Here, 18 equals 2 multiplied by 3 squared ($ 3 \ imes 3 $).", "- 30 breaks down as:\n $ 30 = 2 \ imes 3 \ imes 5 $\n This shows 30 as the product of 2, 3, and 5 — all distinct primes.", "### Why Factorization Matters: Calculating the LCM", "The Least Common Multiple (LCM) of several numbers is the smallest positive number that is divisible by each of them. To find the LCM using prime factorization, follow these steps:", "1. Identify all unique prime factors from the given numbers.\n2. Take the highest power of each prime found in any factorization.\n3. Multiply these highest powers together to get the LCM.", "For example, consider the numbers 12, 18, and 30:", "- The primes involved are: 2, 3, and 5.\n- The highest power of 2 is $ 2^2 $ (from 12).\n- The highest power of 3 is $ 3^2 $ (from 18).\n- The highest power of 5 is $ 5^1 $ (from 30).", "Now multiply these together:\n$$\n\ ext{LCM} = 2^2 \ imes 3^2 \ imes 5 = 4 \ imes 9 \ imes 5 = 180\n$$", "### Final Result", "$$\n12 = 2^2 \ imes 3,\quad 18 = 2 \ imes 3^2,\quad 30 = 2 \ imes 3 \ imes 5\n$$\n$$\n\ ext{LCM}(12, 18, 30) = 2^2 \ imes 3^2 \ imes 5 = 180\n$$", "This method offers an efficient and accurate way to compute the LCM, ensuring clarity and correctness in problem-solving. Understanding factorization unlocks deeper insights into number relationships and simplifies complex calculations in math and daily life."]

Solution: Factorize each number: $ 12 = 2^2 \times 3 $, $ 18 = 2 \times 3^2 $, and $ 30 = 2 \times 3 \times 5 $. The LCM is the product of the highest powers of all primes: $ 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 180 $.

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