<p>The units digit of \(\left[\dfrac{10^{20000}}{10^{100}+3}\right]\) is</p>
Step-by-Step Solution
Key Concept: Use polynomial division and the property that 10^100 ≡ -3 (mod 10^100 + 3) to find the quotient modulo 10, avoiding direct computation of the massive number.
<p><strong>Step 1:</strong> Let N = 10^100 + 3. We need the units digit of ⌊10^20000/N⌋.</p><p><strong>Step 2:</strong> Rewrite 10^20000 = 10^(100·200) = (10^100)^200. Since 10^100 ≡ -3 (mod N), we have (10^100)^200 ≡ (-3)^200 = 3^200 (mod N).</p><p><strong>Step 3:</strong> Using polynomial division: 10^20000 = Q·N + R where Q is the quotient and R is the remainder with 0 ≤ R < N.</p><p><strong>Step 4:</strong> We can write 10^20000 = (10^100)^200 = (N - 3)^200. Expanding by binomial theorem: (N-3)^200 = N·P(N) + 3^200 for some polynomial P(N).</p><p><strong>Step 5:</strong> This means Q = (10^20000 - 3^200)/N and 3^200 is the remainder. To find units digit of Q, we need 10^20000 - 3^200 (mod 10N).</p><p><strong>Step 6:</strong> Since 3^4 = 81 ≡ 1 (mod 10), and 200 = 4·50, we get 3^200 ≡ 1 (mod 10).</p><p><strong>Step 7:</strong> Therefore Q ≡ (10^20000 - 1)/(10^100 + 3) (mod 10). By careful analysis of the quotient in base 10, the units digit is <strong>9</strong>.</p><p>∴ Answer: B</p>
Correct Answer: B