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Home » Stock DK Shock: The Hidden Billion-Dollar Move Every Trader Misses! - Sterling Industries

Stock DK Shock: The Hidden Billion-Dollar Move Every Trader Misses! - Sterling Industries

📅 April 18, 2026 👤 scraface

Apr 18, 2026

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📌 Similarly, test residues modulo 5:

📌 \( n \equiv 0 \pmod{5} \Rightarrow n^2 \equiv 0 \pmod{5} \),

📌 \( n \equiv 1 \pmod{5} \Rightarrow n^2 \equiv 1 \pmod{5} \),

📌 \( n \equiv 2 \pmod{5} \Rightarrow n^2 \equiv 4 \pmod{5} \),

📌 \( n \equiv 3 \pmod{5} \Rightarrow n^2 \equiv 9 \equiv 4 \pmod{5} \),

📌 \( n \equiv 4 \pmod{5} \Rightarrow n^2 \equiv 16 \equiv 1 \pmod{5} \).

📌 So, \( n^2 \equiv 1 \pmod{5} \) when \( n \equiv 1 \) or \( 4 \pmod{5} \).

📌 Step 3: Combine using Chinese Remainder Theorem

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\frac{3.75 + 8.25}{2} = \frac{12}{2} = 6
Thus, the average length is $ \boxed{6} $
Question: How many positive 4-digit numbers are divisible by 11?
Solution: The smallest 4-digit number is 1000, and the largest is 9999...
We find the smallest 4-digit number divisible by 11 by dividing 1000 b...
\left\lceil \frac{1000}{11} \right\rceil = \left\lceil 90.91 \right\rc...
So the smallest 4-digit number divisible by 11 is $ 11 \times 91 = 100...
Now, divide 9999 by 11 to find the largest:
\left\lfloor \frac{9999}{11} \right\rfloor = \left\lfloor 909 \right\r...
So the largest 4-digit number divisible by 11 is $ 11 \times 909 = 999...

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