<p>A natural number \(N\) is chosen randomly from the set \(\{1, 2, 3, \ldots, 1000\}\). The probability that \(N\) is a divisor of 10000 is</p>
Step-by-Step Solution
Key Concept: 10000 = 2^4 \times 5^4. Divisors \leq 1000 have the form 2ᵃ5ᵇ with 0 \leq a,b \leq 4, and 2ᵃ5ᵇ \leq 1000.
<p>\(10000 = 2^4 \times 5^4\). All divisors have the form \(2^a 5^b\), \(0 \leq a, b \leq 4\).</p><p>Divisors \(\leq 1000\): enumerate \(2^a 5^b \leq 1000\):</p><ul><li>\(b=0\): 1,2,4,8,16 (5 values)</li><li>\(b=1\): 5,10,20,40,80 (5)</li><li>\(b=2\): 25,50,100,200,400 (5)</li><li>\(b=3\): 125,250,500,1000 (4)</li><li>\(b=4\): 625 only (1)</li></ul><p>Total \(= 5+5+5+4+1 = 20\).</p><p>\(P = \dfrac{20}{1000} = \dfrac{1}{50}\)</p>
Correct Answer: A