\boxed{2}Question: What is the smallest four-digit number that is a multiple of the least common multiple of the numbers $12$ and $18$, and also divisible by the square of the smallest prime greater than $10$?

["Understanding the Smallest Four-Digit Number Divisible by LCM(12, 18) and the Square of the Smallest Prime Greater Than 10", "If you're curious about number theory puzzles or responsible for solving complex math-based queries—especially those involving multiples and divisibility—this article dives deep into a single but precise mathematical challenge:\nWhat is the smallest four-digit number that is a multiple of the least common multiple (LCM) of 12 and 18, and also divisible by the square of the smallest prime number greater than 10?", "Let’s break this question down step by step to uncover the answer using clear reasoning and strong SEO optimization.", "---", "### Step 1: Find the LCM of 12 and 18", "To solve the first part, we need to compute the least common multiple (LCM) of 12 and 18.\nStart by factoring both numbers:\n- $ 12 = 2^2 \ imes 3 $\n- $ 18 = 2 \ imes 3^2 $", "The LCM takes the highest power of each prime factor:\n- $ \ ext{LCM}(12, 18) = 2^2 \ imes 3^2 = 4 \ imes 9 = 36 $", "So, every number that satisfies the LCM condition must be a multiple of 36.", "---", "### Step 2: Identify the Smallest Prime Greater Than 10", "Prime numbers are natural numbers greater than 1 with no positive divisors other than 1 and themselves.\nThe primes greater than 10 are:\n11, 13, 17, 19, ...\nThus, the smallest prime greater than 10 is 11.", "We now need the square of 11:\n$ 11^2 = 121 $", "---", "### Step 3: Find the Smallest Four-Digit Multiple of Both 36 and 121", "We now seek the smallest four-digit number divisible by both 36 and 121, i.e., divisible by their least common multiple.", "First, check if 36 and 121 are coprime:\n- $ 36 = 2^2 \ imes 3^2 $\n- $ 121 = 11^2 $", "They share no common prime factors, so their LCM is just their product:\n$$\n\ ext{LCM}(36, 121) = 36 \ imes 121 = 4356\n$$", "Now, we need the smallest four-digit multiple of 4356.", "Digit count check:\n- The smallest four-digit number is 1000.\n- Compute $ 4356 \ imes 1 = 4356 $, which is already ≥ 1000 and < 10000 → a four-digit number.", "Hence, 4356 is the smallest four-digit number that is a multiple of both 36 and 121 (hence divisible by LCM(12,18) and $11^2$).", "---", "### Why This Problem Matters: Applications and Learning Value", "Understanding how to compute LCMs and apply divisibility rules is crucial in fields like cryptography, computer science, and algorithm design. This problem exemplifies how combining multiple constraints leads to a precise numerical solution—ideal for students, educators, or anyone exploring mathematical reasoning.", "---", "### Final Answer", "The smallest four-digit number that is a multiple of $ \ ext{LCM}(12, 18) = 36 $ and divisible by the square of the smallest prime greater than 10 ($ 11^2 = 121 $) is:", "4356", "---", "Keywords:\nsmallest four-digit number, LCM of 12 and 18, multiple of 36, smallest prime greater than 10, square of prime 11, LCM(12,18), divisibility problem, math puzzles, number theory, prime factorization", "Meta Description:\nDiscover how to find the smallest four-digit number divisible by $ \ ext{LCM}(12, 18) $ and $ 11^2 $. Learn step-by-step through prime factor analysis, LCM computation, and practical number theory."]