Question: What is the probability that a randomly selected positive integer less than or equal to 50 is a factor of 50?
["Title: Probability That a Random Positive Integer ≤ 50 Is a Factor of 50: A Clear Explanation", "---", "Introduction", "Understanding the probability that a randomly selected positive integer less than or equal to 50 is a factor of 50 is a classic problem in number theory and probability. This article explores how to calculate this probability step by step, explains the mathematical reasoning behind determining factors, and provides practical insight into applying probability concepts to real-world number sets. Whether you're a student learning factors, a teacher illustrating probability, or someone curious about integer divisibility, this guide offers a thorough understanding.", "---", "### What Does It Mean?", "We are asked:\nWhat is the probability that a randomly selected positive integer ( n ) with ( 1 \leq n \leq 50 ) is such that ( n ) divides 50?", "To solve this, we need:", "1. The total number of possible outcomes — integers from 1 to 50.\n2. The number of favorable outcomes — integers in this range that divide 50 evenly.", "This ratio gives us the probability.", "---", "### Step 1: Count the Total Possible Outcomes", "Since we select a positive integer from 1 to 50, inclusive, the total number of possible integers is:", "[\n50 - 1 + 1 = 50\n]", "So, there are 50 equally likely outcomes.", "---", "### Step 2: Find All Positive Factors of 50", "A factor of 50 is a positive integer that divides 50 without leaving a remainder. To list all factors of 50, we begin with its prime factorization.", "[\n50 = 2 \ imes 25 = 2 \ imes 5^2\n]", "The full list of positive factors is obtained by combining powers of the prime factors:", "[\n1, 2, 5, 10, 25, 50\n]", "We verify each:", "- (50 \div 1 = 50) → integer\n- (50 \div 2 = 25) → integer\n- (50 \div 5 = 10) → integer\n- (50 \div 10 = 5) → integer\n- (50 \div 25 = 2) → integer\n- (50 \div 50 = 1) → integer", "No other integers ≤ 50 divide 50 evenly.", "Thus, there are 6 factors of 50.", "---", "### Step 3: Count Favorable Outcomes", "Among the 50 possible integers, only the 6 factors (1, 2, 5, 10, 25, 50) are divisible by 50. Therefore, the number of favorable outcomes is:", "[\n6\n]", "---", "### Step 4: Calculate the Probability", "Probability ( P ) is given by:", "[\nP = \frac{\ ext{Number of favorable outcomes}}{\ ext{Total number of outcomes}} = \frac{6}{50}\n]", "Simplify the fraction:", "[\n\frac{6}{50} = \frac{3}{25}\n]", "As a decimal, this is ( 0.12 ), or 12%.", "---", "### Final Answer", "The probability that a randomly selected positive integer less than or equal to 50 is a factor of 50 is:", "[\n\boxed{\frac{3}{25} \ ext{ or } 12%}\n]", "---", "### Why This Matters", "Understanding factor probabilities helps in probability, number theory, and even algorithm design where divisibility constraints appear. Recognizing how to count factors efficiently — using prime factorization — enables quick computation for larger numbers too.", "---", "Key Takeaways:\n- Total outcomes = 50\n- Favorable outcomes = 6 (factors of 50)\n- Probability = ( \frac{3}{25} )\n- This concept applies broadly to combinatorics and discrete mathematics.", "---", "Keywords:\nprobability of a factor, random integer ≤ 50, factor of 50, divisibility probability, number theory, mathematical probability, finding factors of 50, total outcomes 50, favorable outcomes count", "Meta Description:\nLearn how to calculate the probability that a randomly selected positive integer ≤ 50 is a factor of 50. Step-by-step explanation with prime factorization, total outcomes, factor list, and simplification to ( \frac{3}{25} ).", "---\nRead more about probability with factors, divisibility rules, and integer sets in number theory basics."]
