Solution:** First, determine how many positive integers \( n \) satisfy \( 1 \leq n \leq 60 \) and \( n \mid 120 \).
["How Many Positive Integers ( n ) Between 1 and 60 Divide 120?", "When analyzing divisors of a number, especially in mathematical and competitive contexts, understanding exactly how many positive integers satisfy specific conditions is essential. One such problem asks: How many positive integers ( n ) satisfy ( 1 \leq n \leq 60 ) and ( n \mid 120 )?", "### Understanding the Problem", "We are given:\n- The range: ( 1 \leq n \leq 60 )\n- The condition: ( n ) divides 120 (denoted ( n \mid 120 ))", "We want to count all positive integers ( n ) in this range that are divisors of 120.", "### Step 1: Find All Positive Divisors of 120", "First, factorize 120 into its prime factors:\n[\n120 = 2^3 \ imes 3^1 \ imes 5^1\n]", "Using the formula for counting divisors — if ( n = p_1^{a_1} \ imes p_2^{a_2} \ imes \cdots \ imes p_k^{a_k} ), then the total number of positive divisors is:\n[\n(a_1 + 1)(a_2 + 1)\cdots(a_k + 1)\n]\nThus, the number of positive divisors of 120 is:\n[\n(3+1)(1+1)(1+1) = 4 \ imes 2 \ imes 2 = 16\n]", "So, 120 has 16 positive divisors in total. But not all of them fall within ( n \leq 60 ).", "### Step 2: List the Divisors of 120", "We list all divisors of 120 using combinations of its prime powers:\n[\n\begin{aligned}\n&1, \\n&2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120 \\n\end{aligned}\n]", "Now, exclude any divisor greater than 60. The only divisor of 120 exceeding 60 is 120 itself.", "Therefore, valid divisors satisfying ( 1 \leq n \leq 60 ) are:\n[\n1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60\n]", "### Step 3: Count the Valid Divisors", "Counting the numbers listed above:\nThere are 15 numbers.", "### Conclusion", "Out of all positive integers ( n ) from 1 to 60, 15 of them divide exactly 120.", "This result combines prime factorization with careful enumeration — a powerful approach for problems involving divisors in a bounded range.", "Key takeaway: To count divisors of 120 up to 60, list all divisors (16 total), exclude 120 (which is outside the range), and count the remaining 15.", "Efficiently combining theory and verification ensures accuracy — especially useful for math competitions, number theory problems, or algorithm design.", "---", "Keywords: divisors of 120, count divisors up to 60, number theory, positive integers ( n ), divisor function, prime factorization."]
