Solution: To find the arithmetic mean, we first identify the positive integers less than 30 that are divisible by both 3 and 5. These numbers are multiples of the least common multiple of 3 and 5, which is 15. The multiples of 15 less than 30 are:
["Finding the Arithmetic Mean of Positive Integers Less Than 30 Divisible by Both 3 and 5: A Step-by-Step Solution", "Understanding the arithmetic mean is fundamental in mathematics. But what happens when you need to compute it for a specific set of numbers—such as those positive integers less than 30 that are divisible by both 3 and 5? This article explains how to find the arithmetic mean through a clear and logical process, highlighting key concepts along the way.", "---", "### Step 1: Identify the Relevant Numbers", "To find the arithmetic mean of positive integers less than 30 and divisible by both 3 and 5, we first determine the numbers that meet this condition.", "Since 3 and 5 are co-prime (they share no common factors other than 1), multiples of both are simply multiples of their least common multiple (LCM).", "The least common multiple of 3 and 5 is:", "[\n\ ext{LCM}(3, 5) = 15\n]", "Next, list all positive multiples of 15 that are less than 30:", "[\n15, 30, 45, \dots\n]", "Only 15 satisfies the condition (less than 30), so the relevant numbers are:", "[\n{15}\n]", "---", "### Step 2: Compute the Arithmetic Mean", "The arithmetic mean (average) of a set of numbers is the sum of the numbers divided by how many numbers there are.", "With only one number—15—the sum is simply 15, and the count is 1:", "[\n\ ext{Mean} = \frac{15}{1} = 15\n]", "---", "### Why This Sets the Foundation for Problem Solving", "Although only one number appears in our set, recognizing the underlying pattern using the LCM of 3 and 5 gives powerful insight. This approach scales efficiently to larger sets defined by multiple conditions—such as numbers divisible by 3 and 5, or other arithmetic sequences.", "By identifying the foundation (here, the LCM) and systematically computing the mean, learners build confidence and clarity in applying mathematical reasoning.", "---", "### Summary", "- Key condition: Positive integers less than 30 divisible by both 3 and 5.\n- Basis: These numbers are multiples of LCM(3, 5) = 15.\n- Solution set: {15}\n- Arithmetic mean: 15", "This method combines number theory with basic statistics, offering a clear path to solving similar problems in math education.", "---", "Practice Tip: Try finding the arithmetic mean of other similar sets—like numbers divisible by 4 and 7 below 100. Start by computing the LCM, list valid numbers, then apply the mean formula. Understanding these steps makes arithmetic mean calculation both intuitive and reliable.", "---\nMastering such foundational steps enhances your ability to tackle advanced math problems with confidence."]
