<p>The number of even divisors of the number <em>N</em> = 12600 is</p>
Step-by-Step Solution
Key Concept: An even divisor must contain at least one factor of 2. First find prime factorization of N, then count divisors with the constraint that the power of 2 is at least 1.
<p><strong>Step 1:</strong> Find prime factorization of N = 12600</p><p>12600 = 126 × 100 = (2 × 63) × (4 × 25) = (2 × 9 × 7) × (4 × 25)</p><p>12600 = 2³ × 3² × 5² × 7¹</p><p><strong>Step 2:</strong> Count even divisors</p><p>An even divisor has the form 2^a × 3^b × 5^c × 7^d where:</p><p>• a ∈ {1, 2, 3} (at least one factor of 2)</p><p>• b ∈ {0, 1, 2}</p><p>• c ∈ {0, 1, 2}</p><p>• d ∈ {0, 1}</p><p><strong>Step 3:</strong> Apply counting principle</p><p>Number of even divisors = (3) × (3) × (3) × (2) = 54</p><p>∴ Answer: B</p>
Correct Answer: B