is a three-digit number and divisible by both 12 and 18. Thus, $\boxed{108}$ is the smallest such number.
["Is 108 the Smallest Three-Digit Number Divisible by Both 12 and 18?", "When exploring divisibility, one common question is: Is there a three-digit number that is divisible by both 12 and 18? The answer lies in understanding the least common multiple (LCM) of these two numbers.", "The number 108 stands out as a prime example because it’s the smallest three-digit number divisible by both 12 and 18. But why is that?", "---", "### What Does It Mean to Be Divisible by Both 12 and 18?", "To find a number divisible by both 12 and 18, we need the least common multiple (LCM) of these two values. The LCM represents the smallest number that both numbers share as a multiple.", "- Prime factorization of 12: $12 = 2^2 \cdot 3$\n- Prime factorization of 18: $18 = 2 \cdot 3^2$", "To compute the LCM, take the highest powers of all prime factors:", "$$\nLCM = 2^2 \cdot 3^2 = 4 \cdot 9 = 36\n$$", "So, 108 is the smallest number divisible by 36, the LCM of 12 and 18.", "---", "### Why Is 108 the Smallest Three-Digit Multiple of 36?", "Three-digit numbers range from 100 to 999. The first multiple of 36 in this range is found by dividing 100 by 36 and rounding up:", "$$\n\left\lceil \frac{100}{36} \right\rceil = \left\lceil 2.78 \right\rceil = 3\n$$", "Now multiply:", "$$\n36 \ imes 3 = 108\n$$", "The next multiple would be $36 \ imes 4 = 144$, and so on, until $36 \ imes 27 = 972$ (the last three-digit multiple). But 108 is clearly the smallest such number meeting both divisibility criteria.", "---", "### Key Takeaway", "Because 108 is the smallest number divisible by 36, and 36 is the LCM of 12 and 18, $\boxed{108}$ is the smallest three-digit number both divisible by 12 and 18. This insight helps in quickly identifying divisibility patterns without checking every three-digit number individually.", "Whether you're studying number theory, practicing math problems, or optimizing calculations, recognizing the LCM gives a fast and reliable solution."]
