Since \( v \) is a positive integer \(\leq 30\), there are 30 possible choices for \( v \). Therefore, the probability that \( v \) is a factor of 360 is:

["Title: Probability That a Positive Integer ( v \leq 30 ) Is a Factor of 360: A Simple Counting Approach", "---", "Since ( v ) is a positive integer less than or equal to 30, there are exactly 30 possible values:\n[\nv = 1, 2, 3, \dots, 30\n]\nEach choice of ( v ) is equally likely, so we can analyze how many of these values divide 360 evenly.", "### Step 1: Understanding the Factors of 360\nTo find the probability that a randomly selected ( v \leq 30 ) is a factor of 360, we first list all positive factors of 360.", "Start with the prime factorization of 360:\n[\n360 = 2^3 \ imes 3^2 \ imes 5^1\n]\nThe number of positive factors is found by adding 1 to each exponent and multiplying:\n[\n(3+1)(2+1)(1+1) = 4 \ imes 3 \ imes 2 = 24\n]\nSo, 360 has 24 positive factors in total. We now identify which of these factors are less than or equal to 30.", "### Step 2: List All Factors of 360 That Are ≤ 30\nFrom the complete list of factors (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, ...), the factors (\leq 30) are:\n[\n1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30\n]\nCounting these, we find there are 15 such values.", "### Step 3: Compute the Probability\nThere are 30 equally likely choices for ( v ), and 15 of them are factors of 360. Therefore, the probability is:\n[\n\frac{15}{30} = \frac{1}{2}\n]", "### Conclusion\nSince ( v ) is a positive integer ( \leq 30 ), and there are 30 choices, with 15 of those being factors of 360, the probability that a randomly selected ( v ) is a factor of 360 is exactly ( \frac{1}{2} ). This demonstrates how simple enumeration and factor counting yield accurate probability results without advanced statistics.", "Keywords: probability, positive integers, factors of 360, v ≤ 30, factor probability, combinatorics, math teaching, elementary number theory.", "---", "Whether you're a student sampling integers or a math enthusiast exploring divisibility, understanding factor relationships within a bounded set provides clear insight into probability—rooted in counting and divisible relationships.", "For more on factor analysis and probability, explore our guides on counting divisors, factor trees, and numerical probability applications."]