<p>Find the number of factors of the number 37800. Also find the sum of the odd proper divisors of the number.</p>
Step-by-Step Solution
Key Concept: Prime factorize 37800 completely, then use the formula: if n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ, the number of divisors is (a₁+1)(a₂+1)...(aₖ+1). For odd divisors, exclude all factors of 2 and apply the sum formula (p₁^0 + p₁^1 + ... + p₁^a₁)/(p₁-1).
<p><strong>Step 1: Prime Factorization</strong></p><p>37800 = 378 × 100 = 2 × 189 × 100 = 2 × 27 × 7 × 4 × 25 = 2³ × 3³ × 5² × 7¹</p><p><strong>Step 2: Number of Factors</strong></p><p>Using the divisor formula: If n = p₁^a₁ × p₂^a₂ × ... × pₖ^aₖ, then τ(n) = (a₁+1)(a₂+1)...(aₖ+1)</p><p>Number of factors = (3+1)(3+1)(2+1)(1+1) = 4 × 4 × 3 × 2 = <strong>96</strong></p><p><strong>Step 3: Sum of Odd Proper Divisors</strong></p><p>Odd divisors come from odd prime factors only (excluding 2): 3³ × 5² × 7¹</p><p>Sum of all odd divisors = (3⁰ + 3¹ + 3² + 3³)(5⁰ + 5¹ + 5²)(7⁰ + 7¹)</p><p>= (1 + 3 + 9 + 27)(1 + 5 + 25)(1 + 7)</p><p>= (40)(31)(8) = 9920</p><p><strong>Step 4: Proper Divisors Adjustment</strong></p><p>Proper divisors exclude the number itself. The largest odd divisor is 3³ × 5² × 7 = 27 × 25 × 7 = 4725</p><p>Sum of odd proper divisors = 9920 - 1 = <strong>9919</strong></p><p>(Note: We subtract 1, not 4725, because the number 37800 itself is not odd; the sum formula already excludes it implicitly)</p>
Correct Answer: Number of factors = 96, Sum of odd proper divisors = (3^0 + 3^1 + 3^2 + 3^3)(5^0 + 5^1 + 5^2)(7^0 + 7^1) - 1