<p>Find the number of divisors of 720. How many of these are even? Also find the sum of divisors.</p>
Step-by-Step Solution
Key Concept: Prime factorize 720 = 2⁴ × 3² × 5¹, then use the formula: number of divisors = (4+1)(2+1)(1+1) and sum of divisors = (2⁵-1)/(2-1) × (3³-1)/(3-1) × (5²-1)/(5-1). Even divisors exclude the case where power of 2 is 0.
<p><strong>Step 1: Prime Factorization</strong></p><p>720 = 16 × 45 = 2⁴ × 3² × 5¹</p><p><strong>Step 2: Total Number of Divisors</strong></p><p>Using the formula: τ(n) = (a₁+1)(a₂+1)(a₃+1) where n = p₁^a₁ × p₂^a₂ × p₃^a₃</p><p>τ(720) = (4+1)(2+1)(1+1) = 5 × 3 × 2 = <strong>30 divisors</strong></p><p><strong>Step 3: Number of Even Divisors</strong></p><p>Even divisors must contain at least 2¹. So we take 2¹ to 2⁴ (4 choices), all powers of 3 (3 choices), and all powers of 5 (2 choices)</p><p>Even divisors = (4)(3)(2) = <strong>24</strong></p><p><strong>Step 4: Sum of Divisors</strong></p><p>σ(720) = σ(2⁴) × σ(3²) × σ(5¹)</p><p>σ(2⁴) = (2⁵-1)/(2-1) = 31</p><p>σ(3²) = (3³-1)/(3-1) = 26/2 = 13</p><p>σ(5¹) = (5²-1)/(5-1) = 24/4 = 6</p><p>σ(720) = 31 × 13 × 6 = 403 × 6 = <strong>2418</strong></p>
Correct Answer: 30 divisors, sum of divisors = 2418