Solution: We seek the smallest three-digit number divisible by both 12 and 18, i.e., their least common multiple.
["Solution to Finding the Smallest Three-Digit Number Divisible by Both 12 and 18 (Their Least Common Multiple)", "When tackling number theory problems, identifying the least common multiple (LCM) is crucial, especially when working with divisibility requirements. In this article, we explore how to determine the smallest three-digit number divisible by both 12 and 18—essentially finding their LCM and then locating the smallest such three-digit value.", "### Understanding the Least Common Multiple (LCM)", "The least common multiple of two or more numbers is the smallest positive integer that is evenly divisible by each of them. For this problem, we need the LCM of 12 and 18.", "Why find LCM?\nLCM helps in solving problems involving schedules, repetitions, and synchronization. For practical use, this helps identify numbers shared across multiple divisors—especially useful in math problems, engineering, and computer science.", "### Step 1: Factorize the Numbers", "To compute the LCM accurately, factor each number into its prime factors.", "- ( 12 = 2^2 \ imes 3 )\n- ( 18 = 2 \ imes 3^2 )", "### Step 2: Take the Highest Powers of All Primes", "The LCM is obtained by multiplying the highest powers of all prime factors present.", "- The highest power of 2 is ( 2^2 )\n- The highest power of 3 is ( 3^2 )", "So,", "[\n\ ext{LCM}(12, 18) = 2^2 \ imes 3^2 = 4 \ imes 9 = 36\n]", "### Step 3: Find the Smallest Three-Digit Multiple of 36", "Now, we need the smallest number ≥ 100 that is divisible by 36.", "We start by dividing 100 by 36 to determine the next multiple:", "[\n100 \div 36 \approx 2.78\n]", "The smallest whole number greater than or equal to 2.78 is 3. Multiply 36 by 3:", "[\n36 \ imes 3 = 108\n]", "### Step 4: Verify", "Check that 108 is a three-digit number and divisible by both 12 and 18:", "- ( 108 \div 12 = 9 ) → divisible by 12\n- ( 108 \div 18 = 6 ) → divisible by 18\n- ( 108 \div 36 = 3 ) → divisible by LCM", "### Conclusion", "The smallest three-digit number divisible by both 12 and 18—in other words, their least common multiple—is 108. This solution demonstrates a clear method to compute LCM and identify relevant multiples efficiently, helping solve real-world and mathematical problems involving common multiples.", "---", "Keywords:\nsmallest three-digit number divisible by 12 and 18, LCM of 12 and 18, least common multiple solution, math problem, number theory, find LCM, 12 and 18 LCM, divisibility examples, practical math solution", "Meta Description:\nDiscover how to compute the smallest three-digit number divisible by both 12 and 18 using prime factorization and LCM. Learn step-by-step how to solve this common number theory problem."]
