Solution: Find the LCM of 12 and 18. Prime factors: $12 = 2^2 imes 3$, $18 = 2 imes 3^2$. LCM is $2^2 imes 3^2 = 36$. The smallest three-digit multiple of 36 is $36 imes 3 = 108$. The population threshold is $oxed{108}$.
["How to Find the LCM of 12 and 18: Prime Factorization & Smallest Three-Digit Multiple", "Understanding how to calculate the Least Common Multiple (LCM) is essential in math and everyday problem-solving. In this article, we’ll walk through finding the LCM of 12 and 18 using prime factorization, and discover the smallest three-digit number that is a multiple of their LCM.", "### Understanding LCM", "The LCM of two or more numbers is the smallest positive integer that is divisible by each of them. For example, finding the LCM of 12 and 18 helps us determine the smallest number we can divide evenly by both.", "### Prime Factorization Method", "To compute the LCM using prime factorization:", "1. Break down each number into prime factors:\n - $12 = 2^2 \ imes 3^1$\n - $18 = 2^1 \ imes 3^2$", "2. Take the highest power of each prime number:\n - For $2$: highest power is $2^2$\n - For $3$: highest power is $3^2$", "3. Multiply these together to get the LCM:\n $$\n \ ext{LCM}(12, 18) = 2^2 \ imes 3^2 = 4 \ imes 9 = 36\n $$", "### Finding the Smallest Three-Digit Multiple of 36", "Now that we know $LCM(12, 18) = 36$, we want the smallest three-digit number divisible by 36.", "Three-digit numbers start from $100$. To find the smallest multiple of 36 that is at least 100:", "- Divide $100$ by $36$:\n $$\n 100 \div 36 \approx 2.78\n $$\n- Round up to the next whole number: $3$\n- Multiply: $36 \ imes 3 = 108$", "So, the smallest three-digit multiple of 36—and the population threshold— is $108$.", "### Summary", "- LCM of $12$ and $18$: $36$\n- Smallest three-digit multiple of $36$: $108$\n- This number marks a key threshold in counting or grouping tasks involving these two values.", "Use this step-by-step method whenever you need to calculate LCMs or locate multiples in math problems!", "---", "\boxed{108} — the smallest three-digit multiple of the LCM of 12 and 18."]
