Solution: First, we determine all positive integers less than or equal to 50 that are factors of 50.
["Solution: Identifying All Positive Integers Less Than or Equal to 50 That Are Factors of 50", "Understanding mathematical factors is essential in number theory and problem-solving. When asked to determine all positive integers less than or equal to 50 that are factors of 50, we begin by identifying which whole numbers divide 50 evenly without leaving a remainder.", "### Step 1: Recognize the Definition of a Factor\nA factor of a number is any integer that divides that number exactly—meaning there is no remainder. To solve this problem, we examine all integers from 1 to 50 and check if they divide 50 evenly.", "### Step 2: Prime Factorization of 50\nBreaking 50 into its prime components helps clarify its divisors:\n[\n50 = 2 \ imes 5^2\n]\nThis prime factorization reveals that the base prime factors are 2 and 5, raised to powers of 1 and 2, respectively.", "### Step 3: List All Positive Factors\nUsing the prime factors, we generate all possible divisors by combining powers of 2 and 5. The complete set of positive integer factors of 50 is:\n- (1) (since (1 \ imes 50 = 50))\n- (2) (from prime factor 2)\n- (5) (from prime factor 5)\n- (10 = 2 \ imes 5)\n- (25 = 5^2)\n- (50 = 2 \ imes 5^2)", "### Step 4: Confirm All Factors Are ≤ 50\nBy definition, all the factors listed—1, 2, 5, 10, 25, and 50—are less than or equal to 50, so they fully satisfy the condition.", "### Summary of Valid Factors:\n[\n\ ext{Factors of 50 that are } \leq 50: {1, 2, 5, 10, 25, 50}\n]", "This methodical approach ensures accuracy and clarity when identifying integer factors within a specified range. Recognizing factors not only aids in basic arithmetic but also supports deeper understanding in algebra, divisibility rules, and factor trees.", "### Why This Matters\nKnowing the factors of a number helps in simplifying fractions, solving equations, and analyzing the structure of numbers. Whether in classroom learning or practical applications, identifying factors within limits like 50 supports efficient computation and conceptual mastery.", "---", "By systematically listing and verifying all divisors, we confirm that the positive integers less than or equal to 50 and factors of 50 are clearly characterized, providing a strong foundation for further mathematical exploration."]
