Solution: First, factor $ 180 = 2^2 \cdot 3^2 \cdot 5 $. To find only the odd divisors, we ignore the factor of 2 and consider only $ 3^2 \cdot 5 $.
["Solution: Finding Odd Divisors of 180 Using Prime Factorization", "When tasked with finding only the odd divisors of a number, it's essential to eliminate even factors. Let’s explore a clear and efficient mathematical method using prime factorization.", "---", "### Step 1: Prime Factorization of 180", "Begin by expressing 180 in its complete prime factorized form:\n$$\n180 = 2^2 \cdot 3^2 \cdot 5\n$$", "---", "### Step 2: Understanding Why to Ignore $ 2^2 $", "The factor 2 appears to the power of 2, contributing even divisors when combined with any other primes. However, for odd divisors, even factors must be excluded. Thus, we focus only on the odd part of the factorization:\n$$\n3^2 \cdot 5\n$$", "---", "### Step 3: How to Find All Odd Divisors", "To find all divisors of the odd part $ 3^2 \cdot 5 $, we apply the classic rule:\nFor any prime factor expressed as $ p^k $, the number of divisors contributed is $ k+1 $. Multiplying the possible exponents, the total number of divisors is $ (2+1)(1+1) = 6 $.", "Now list each combination:\n- $ 3^0 \cdot 5^0 = 1 $\n- $ 3^1 \cdot 5^0 = 3 $\n- $ 3^2 \cdot 5^0 = 9 $\n- $ 3^0 \cdot 5^1 = 5 $\n- $ 3^1 \cdot 5^1 = 15 $\n- $ 3^2 \cdot 5^1 = 45 $", "Thus, the odd divisors of 180 are:\n$$\n\boxed{1,\ 3,\ 5,\ 9,\ 15,\ 45}\n$$", "---", "### Why This Method Works", "By removing all powers of 2 from the prime factorization and computing divisors from the remaining odd prime factors, we efficiently isolate all odd divisors. This technique applies not only to 180 but to any integer—make sure to strip away even components before applying divisor formulas.", "---", "### Summary", "- Prime factorize: $ 180 = 2^2 \cdot 3^2 \cdot 5 $\n- Ignore $ 2^2 $ for odd divisors\n- Apply divisor rule to $ 3^2 \cdot 5 $\n- Compute all combinations to derive the full list of odd divisors: $ 1,\ 3,\ 5,\ 9,\ 15,\ 45 $", "Use this systematic approach for clean, accurate results every time.", "---", "Keywords: find odd divisors of 180, prime factorization method, odd divisors of 180, solution for odd divisors, factor 180 into primes, 3²·5 odd part, divisor calculation, math tutorial, number theory."]
