<p>In how many ways the number 7056 can be resolved as a product of 2 factors?</p>
Step-by-Step Solution
Key Concept: First factorize 7056 completely, then recognize that each divisor of 7056 pairs with another divisor (d, 7056/d) to form the product. The number of ways equals the number of divisors, but we must account for whether the number is a perfect square to avoid double counting unordered pairs.
<p><strong>Step 1:</strong> Prime factorize 7056</p><p>7056 = 16 × 441 = 2⁴ × 3² × 7²</p><p><strong>Step 2:</strong> Find total number of divisors using τ(n) = (4+1)(2+1)(2+1) = 5 × 3 × 3 = 45</p><p><strong>Step 3:</strong> Recognize that 7056 = 84² is a perfect square (√7056 = 84)</p><p><strong>Step 4:</strong> Each divisor d pairs with 7056/d. For unordered pairs (considering 7056 = d₁ × d₂ where d₁ ≤ d₂), count: (τ(n) + 1)/2 for perfect squares (the middle divisor 84 pairs with itself once)</p><p>Number of ways = (45 + 1)/2 = 46/2 = 23</p><p>∴ Answer: <strong>23</strong></p>
Correct Answer: 23