First, we determine the number of positive integers \( v \) that are factors of 360 and are less than or equal to 30. Begin by finding the prime factorization of 360:

["# Finding All Positive Integer Factors of 360 Less Than or Equal to 30", "Understanding how to identify factors efficiently is a fundamental skill in number theory and everyday mathematics. One common problem involves determining how many positive integers ( v ) are factors of 360 and also satisfy ( v \leq 30 ). To solve this, we begin with the prime factorization of 360, which unlocks the full set of its factors.", "## Step 1: Find the Prime Factorization of 360", "To uncover all factors systematically, we start by breaking down 360 into its prime components.", "360 is an even number, so divide by 2:\n360 ÷ 2 = 180\n180 ÷ 2 = 90\n90 ÷ 2 = 45", "So far, we have ( 360 = 2^3 \ imes 45 ).\nNow factor 45: divisible by 3,\n45 ÷ 3 = 15\n15 ÷ 3 = 5\n5 is a prime.", "Putting it all together:\n[\n360 = 2^3 \ imes 3^2 \ imes 5^1\n]", "## Step 2: Generate All Positive Factors of 360", "Using the prime factorization ( 2^3 \ imes 3^2 \ imes 5^1 ), every positive factor of 360 is formed by choosing powers of 2 from ( 2^0 ) to ( 2^3 ), powers of 3 from ( 3^0 ) to ( 3^2 ), and powers of 5 from ( 5^0 ) to ( 5^1 ).", "This means:\n- Powers of 2: ( 2^0 = 1 ), ( 2^1 = 2 ), ( 2^2 = 4 ), ( 2^3 = 8 )\n- Powers of 3: ( 3^0 = 1 ), ( 3^1 = 3 ), ( 3^2 = 9 )\n- Power of 5: ( 5^0 = 1 ), ( 5^1 = 5 )", "Multiplying combinations of these gives all 24 factors of 360 (since the total number of positive factors is ( (3+1)(2+1)(1+1) = 4 \ imes 3 \ imes 2 = 24 )).", "## Step 3: Extract Factors Less Than or Equal to 30", "Now we filter and list only those factors of 360 that are ( \leq 30 ). We systematically combine the prime powers within limits.", "### List all combinations (avoiding duplicates):\n- From ( 2^a \ imes 3^b \ imes 5^c ), where ( a = 0,1,2,3 ), ( b = 0,1,2 ), ( c = 0,1 )", "Start with ( 5^0 = 1 ):\n- ( 1 \ imes 1 = 1 )\n- ( 1 \ imes 2 = 2 )\n- ( 1 \ imes 4 = 4 )\n- ( 1 \ imes 8 = 8 )\n- ( 1 \ imes 3 = 3 )\n- ( 1 \ imes 6 = 6 )\n- ( 1 \ imes 12 = 12 )\n- ( 1 \ imes 24 = 24 )\n- ( 1 \ imes 9 = 9 )\n- ( 1 \ imes 18 = 18 )\n- ( 1 \ imes 36 = 36 ) → too big\n- ( 1 \ imes 5 = 5 )\n- ( 1 \ imes 10 = 10 )\n- ( 1 \ imes 20 = 20 )\n- ( 1 \ imes 40 = 40 ) → too big\n- ( 1 \ imes 15 = 15 )\n- ( 1 \ imes 30 = 30 )", "Now ( 5^1 = 5 ):\n- ( 5 \ imes 1 = 5 ) (already counted)\n- ( 5 \ imes 2 = 10 )\n- ( 5 \ imes 3 = 15 )\n- ( 5 \ imes 4 = 20 )\n- ( 5 \ imes 6 = 30 )\n- ( 5 \ imes 8 = 40 ) → too big\n- ( 5 \ imes 9 = 45 ) → too big\n- ( 5 \ imes 12 = 60 ) → too big\n- ( 5 \ imes 18 = 90 ) → too big", "Next, ( 5^1 \ imes 2^2 = 5 \ imes 4 = 20 ) — already listed\n( 5^1 \ imes 2^3 = 5 \ imes 8 = 40 ) — too big", "Now check ( 5^1 \ imes 3^2 = 5 \ imes 9 = 45 ) — too big", "So collect all unique values ( \leq 30 ):\n1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30", "Double-check completeness:\n- From ( 2^3 \ imes 3 \ imes 5 = 8 \ imes 3 \ imes 5 = 120 ) → too big\n- ( 2^2 \ imes 3^2 = 4 \ imes 9 = 36 ) → too big\n- ( 2 \ imes 3 \ imes 5 = 30 ) — included\n- ( 3^2 \ imes 5 = 45 ) → too big", "Thus, the complete list of positive factors of 360 that are ( \leq 30 ) is:\n[\n{1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30}\n]", "### Count the Valid Factors:\nThere are 15 such numbers.", "## Conclusion", "By starting with prime factorization, systematically generating factor combinations, and filtering by the upper bound, we efficiently determined that 15 positive integers are factors of 360 and less than or equal to 30. This method is both powerful and scalable for similar problems involving large numbers and constraints.", "Whether you're solving math problems, preparing for exams, or just exploring number patterns, mastering factorization and filtering techniques allows faster, more accurate results.", "If you're interested in how many factors of any number meet certain bounds, use prime factorization as your foundation—then systematically explore combinations within limits. This approach is key to mastering number theory and practical problem-solving alike.", "---\nKeywords: factors of 360, prime factorization, number theory, finding factors ≤ 30, divisors of 360, positive integers, factor combinations, math problem solving"]