Solution: The least common multiple (LCM) of 14 and 21 is $42$. The smallest three-digit multiple of 42 is $42 \times 3 = 126$.

["Understanding the Least Common Multiple (LCM) of 14 and 21 and Finding the Smallest Three-Digit Multiple", "When solving problems involving multiples in math, one key concept is the Least Common Multiple (LCM). The LCM of two or more numbers is the smallest positive number that is evenly divisible by each of them. A common example used in basic number theory is the LCM of 14 and 21, a question often explored to build foundational understanding.", "The LCM of 14 and 21 can be calculated using prime factorization:", "- $14 = 2 \ imes 7$\n- $21 = 3 \ imes 7$", "The LCM is obtained by taking the highest power of each prime factor involved:", "$$\n\ ext{LCM}(14, 21) = 2^1 \ imes 3^1 \ imes 7^1 = 42\n$$", "So, the least common multiple of 14 and 21 is 42 — a small yet powerful number in divisibility and fraction operations.", "Understanding this LCM opens the door to finding multiples, especially the smallest three-digit multiple. For instance, knowing that 42 is the LCM helps identify its repeated multiples efficiently. The smallest three-digit number is 100, and to find the smallest three-digit multiple of 42, we divide 100 by 42:", "$$\n100 \div 42 \approx 2.38\n$$", "Since we need the smallest integer multiple that results in a three-digit number, we round up to the next whole number, which is 3. Multiplying:", "$$\n42 \ imes 3 = 126\n$$", "Thus, 126 is the smallest three-digit multiple of 42.", "Wrapping up\nThe LCM of 14 and 21 is $42$, a fundamental number for learning divisibility and shared multiples. From here, identifying the smallest three-digit multiple—like calculating $42 \ imes 3 = 126$—demonstrates practical application. Whether in homework, testing, or real-world math, mastering LCMs provides a strong foundation for better problem-solving.", "---", "Key takeaway:\n- LCM of 14 and 21 = 42\n- The smallest three-digit multiple of 42 = $42 \ imes 3 = 126$\n- Understanding multiples starts with LCMs and builds essential skills in number theory.", "Tags: LCM 14 and 21, smallest three-digit multiple, 42 × 3 = 126, number theory, maths education", "---", "Explore how LCM helps unlock patterns in numbers — perfect for students, teachers, or anyone interested in math fundamentals."]