<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 power of 5. The divisors of 42 give us all possible integer values of the logarithm.
<p><strong>Step 1:</strong> For log_p(5^42) to be an integer, let log_p(5^42) = k where k is a positive integer.</p><p><strong>Step 2:</strong> This means p^k = 5^42, so p = 5^(42/k).</p><p><strong>Step 3:</strong> For p to be a positive integer, (42/k) must be a non-negative integer, meaning k must be a divisor of 42.</p><p><strong>Step 4:</strong> Find divisors of 42: 42 = 2 × 3 × 7, so divisors are {1, 2, 3, 6, 7, 14, 21, 42}.</p><p><strong>Step 5:</strong> For each divisor k of 42:</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 6:</strong> Product = 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