Question:** A statistician is modeling data points that follow a discrete probability distribution. What is the probability that a positive integer less than or equal to 60 is a divisor of 120?

["Probability That a Positive Integer ≤ 60 Is a Divisor of 120: A Statistical Model", "When analyzing real-world data using discrete probability distributions, one common task is determining the likelihood that a randomly selected integer within a given range divides a fixed number—in this case, 120. This article explores the statistical modeling behind the question: What is the probability that a randomly chosen positive integer less than or equal to 60 is a divisor of 120?", "---", "### Understanding the Problem", "We are interested in counting how many positive integers ≤ 60 divide 120, then using this count to compute a probability. Since divisors of 120 are evenly spaced in theory, most lie within the range of 1 to 120, but not all divisors fall below or equal to 60. The probability asks for the ratio:", "> Probability = (Number of positive integers ≤ 60 that divide 120) ÷ 60", "This approach reflects a discrete uniform probability model, where each integer in [1, 60] has an equal chance of being the divisor under consideration.", "---", "### Step 1: Find All Divisors of 120", "First, factor 120 into its prime components:\n120 = 2³ × 3 × 5", "The total number of positive divisors of 120 is given by multiplying one more than each exponent:\n(3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16 divisors.", "We now list all of them:", "1,\n2,\n3,\n4,\n5,\n6,\n8,\n10,\n12,\n15,\n20,\n24,\n30,\n40,\n60,\n120", "---", "### Step 2: Select Divisors ≤ 60", "From the complete list, all divisors except 120 are ≤ 60. Thus, we exclude 120 and count the remaining 15.", "So, there are 15 positive integers ≤ 60 that divide 120.", "---", "### Step 3: Compute the Probability", "Using the discrete uniform model:", "[\n\ ext{Probability} = \frac{\ ext{Number of favorable outcomes}}{\ ext{Total possible outcomes}} = \frac{15}{60} = \frac{1}{4} = 0.25\n]", "---", "### Statistical Interpretation", "This problem exemplifies how probability distributions over discrete sets allow us to quantify the likelihood of structural relationships—in this case, divisibility within a bounded interval. Such modeling is foundational in areas like combinatorial statistics, cryptography (e.g., divisor-based RSA analogs), and probabilistic algorithms.", "Moreover, while the full set of divisors spans up to 120, restricting to ≤ 60 alters the distribution, increasing the probability compared to if the full set were considered. This illustrates how domain constraints shape empirical probabilities.", "---", "### Conclusion", "The probability that a randomly selected positive integer ≤ 60 is a divisor of 120 is exactly 0.25 or 25%. This result emerges cleanly from prime factorization and unbiased selection within a uniform discrete distribution. For statisticians modeling discrete randomness, this example reinforces core concepts in probability theory and distribution analysis.", "---", "Keywords: probability of divisors, discrete probability, statistical modeling, divisor count, uniform distribution, integer divisors, 120 divisors, combinatorics, probability theorem", "Meta Description:\nA statistical model shows that the probability a positive integer ≤ 60 divides 120 is exactly 25%. Learn how divisor enumeration and uniform distribution yield this result."]