<p>The number of integral values of \(p\) such that \(\log_p 5^{42}\) is an integer equals the number of divisors of 42. Find the product of all integral values of \(p\).</p>
Step-by-Step Solution
Key Concept: For log_p(5^42) to be an integer, we need 5^42 = p^k for some integer k, which means p must be a power of 5. The constraint that log_p(5^42) is an integer forces p = 5^d where d divides 42, making the count of valid p values equal to the divisor count of 42.
<p><strong>Step 1:</strong> For log_p(5^42) to be an integer, let log_p(5^42) = k where k ∈ ℤ. Then p^k = 5^42.</p><p><strong>Step 2:</strong> For p to be an integer base with p^k = 5^42, p must be of the form p = 5^(42/k). For p to be an integer, k must divide 42.</p><p><strong>Step 3:</strong> If k divides 42, then p = 5^(42/k). The divisors of 42 are: 1, 2, 3, 6, 7, 14, 21, 42 (total of 8 divisors, which matches the given condition).</p><p><strong>Step 4:</strong> The corresponding values of p are:</p><ul><li>k = 1 → p = 5^42</li><li>k = 2 → p = 5^21</li><li>k = 3 → p = 5^14</li><li>k = 6 → p = 5^7</li><li>k = 7 → p = 5^6</li><li>k = 14 → p = 5^3 = 125</li><li>k = 21 → p = 5^2 = 25</li><li>k = 42 → p = 5^1 = 5</li></ul><p><strong>Step 5:</strong> Product of all integral values = 5^42 × 5^21 × 5^14 × 5^7 × 5^6 × 5^3 × 5^2 × 5^1 = 5^(42+21+14+7+6+3+2+1) = 5^96</p><p>∴ Answer: C</p>
Correct Answer: C