<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>$b=0$: 1,2,4,8,16 (5 values)$b=1$: 5,10,20,40,80 (5)$b=2$: 25,50,100,200,400 (5)$b=3$: 125,250,500,1000 (4)$b=4$: 625 only (1)</ul><p>Total $= 5+5+5+4+1 = 20$.</p><p>$P = \dfrac{20}{1000} = \dfrac{1}{50}$</p>
Correct Answer: A