Question: What is the probability that a randomly chosen positive integer less than or equal to 50 is a factor of 60?
["# What Is the Probability That a Randomly Chosen Positive Integer ≤ 50 Is a Factor of 60?", "When exploring number theory and probability, one intriguing question arises: What is the probability that a randomly chosen positive integer less than or equal to 50 is a factor of 60? This query combines basic number properties with probability, making it an engaging and educational topic for math enthusiasts and students alike.", "## Understanding the Problem", "To solve this, we need two key pieces of information:", "1. All positive integers from 1 to 50 — a total of 50 possible choices.\n2. Which of these integers are factors of 60 — meaning numbers that divide 60 without leaving a remainder.", "The probability is then calculated as:\n[\n\ ext{Probability} = \frac{\ ext{Number of favorable outcomes}}{\ ext{Total number of possible outcomes}} = \frac{\ ext{# factors of 60 ≤ 50}}{50}\n]", "---", "## Step 1: Find All Factors of 60", "First, determine all positive factors of 60. Start with the prime factorization of 60:\n[\n60 = 2^2 \ imes 3 \ imes 5\n]\nUsing the formula for finding the total number of positive factors:\nIf ( n = p_1^{e_1} \ imes p_2^{e_2} \ imes \cdots \ imes p_k^{e_k} ), the number of positive factors is ((e_1+1)(e_2+1)\cdots(e_k+1)).\nSo for 60:\n[\n(2+1)(1+1)(1+1) = 3 \ imes 2 \ imes 2 = 12 \ ext{ total factors}\n]", "Now list all 12 factors of 60:\n[\n1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60\n]", "---", "## Step 2: Identify Which Factors Are ≤ 50", "From the full list of factors, exclude any greater than 50. Only 60 exceeds 50, so we exclude it. The relevant factors are:\n[\n1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30\n]\nThis gives 11 favorable outcomes.", "---", "## Step 3: Calculate the Probability", "Now plug into the probability formula:\n[\n\ ext{Probability} = \frac{\ ext{# factors of 60 ≤ 50}}{\ ext{Total numbers from 1 to 50}} = \frac{11}{50}\n]", "---", "## Final Answer", "The probability that a randomly selected positive integer less than or equal to 50 is a factor of 60 is:\n[\n\boxed{\frac{11}{50}}\n]\nor 22% in percentage form.", "---", "## Why This Matters", "This problem illustrates how fundamental concepts of divisibility integrate with probability. Understanding the distribution of factors helps in number theory, cryptography, and even coding algorithms where efficient checking of divisibility is crucial.", "Whether you're a student preparing for math exams or simply a curious learner, this question beautifully connects arithmetic with probabilistic thinking—proving that even simple queries can open doors to deeper mathematical insight.", "---", "Keywords: probability, positive integers ≤ 50, factor of 60, number theory, divisibility, math problem, probability probability, factor count probability calculation."]
