<p>The product of all positive integral values of \(p\) for which \(\log_p 5^{42}\) is an integer, is:</p>
Step-by-Step Solution
Key Concept: For log_p(5^42) to be an integer k, we need p^k = 5^42, which means p must be a divisor of 5^42. Since p is a positive integer and p^k = 5^42, p can only have 5 as its prime factor, so p = 5^m where m divides 42.
<p><strong>Step 1:</strong> For log_p(5^42) = k (integer), we need p^k = 5^42</p><p><strong>Step 2:</strong> Since p is a positive integer and p^k = 5^42, p must be of the form 5^m where m is a positive integer</p><p><strong>Step 3:</strong> From p^k = 5^42: (5^m)^k = 5^42, so mk = 42</p><p><strong>Step 4:</strong> For positive integer p, we need m ∈ {1, 2, 3, 6, 7, 14, 21, 42} (all positive divisors of 42)</p><p><strong>Step 5:</strong> The corresponding values of p are: 5^1, 5^2, 5^3, 5^6, 5^7, 5^14, 5^21, 5^42</p><p><strong>Step 6:</strong> Product = 5^1 · 5^2 · 5^3 · 5^6 · 5^7 · 5^14 · 5^21 · 5^42 = 5^(1+2+3+6+7+14+21+42) = 5^96</p><p>∴ Answer: D</p>
Correct Answer: D