<p>The probability that a random chosen three-digit number has exactly 3 factors is</p>
Step-by-Step Solution
Key Concept: A number has exactly 3 factors if and only if it is the square of a prime number (since factors of p² are 1, p, and p² only). Find all three-digit perfect squares of primes within the range [100, 999].
<p><strong>Step 1:</strong> A number n has exactly 3 factors ⟺ n = p² where p is prime (factors are 1, p, p²)</p><p><strong>Step 2:</strong> Find three-digit perfect squares of primes in [100, 999]:</p><p>• 11² = 121 ✓</p><p>• 13² = 169 ✓</p><p>• 17² = 289 ✓</p><p>• 19² = 361 ✓</p><p>• 23² = 529 ✓</p><p>• 29² = 841 ✓</p><p>• 31² = 961 ✓</p><p>• 37² = 1369 (four digits, stop)</p><p><strong>Step 3:</strong> Total such numbers = 7</p><p><strong>Step 4:</strong> Total three-digit numbers = 900 (from 100 to 999)</p><p><strong>Step 5:</strong> Probability = 7/900</p><p>∴ Answer: B</p>
Correct Answer: B