Question: Find the smallest positive integer whose square ends in $001$.
["Find the Smallest Positive Integer Whose Square Ends in $001 \nDiscover the Math Behind a Digital Curiosity Driving Online Dialogue", "Is there a number so uniquely tied to the end of its square that it sparks quiet fascination across forums, search bars, and mobile devices? The question Find the smallest positive integer whose square ends in $001 is quietly trending as a blend of number theory and digital curiosity. While not a mainstream topic, it reflects growing interest in modular arithmetic, last digits patterns, and mathematical puzzles—areas increasingly explored in today’s data-driven culture.", "Why This Question Is Gaining Attention in the US", "This inquiry taps into a broader trend: user-driven exploration of math problems online, amplified by mobile search habits and a hunger for intellectual puzzles. In the United States, interest in numerical patterns has grown alongside digital literacy, especially among users seeking quick intellectual rewards. The specificity of “ends in $001” grounds the query in tangible数字 details, making it both precise and accessible. With rising curiosity about personal data patterns, cryptography basics, and digital trends, finding such numbers becomes a micro-adventure in logic—hidden in plain sight within mobile search queries.", "How It Actually Works: A Clear Explanation", "To find the smallest positive integer \( n \) such that \( n^2 \mod 1000 = 1 \), we solve:", "\[\nn^2 \equiv 1 \pmod{1000}\n\]", "This means \( n^2 - 1 \) must be divisible by 1000, or: \n\( n^2 - 1 = (n - 1)(n + 1) \equiv 0 \pmod{1000} \)", "Since \( 1000 = 8 \ imes 125 \), we analyze modulo 8 and 125 separately, then combine results using the Chinese Remainder Theorem.", "Modulo 8: \nSquares mod 8 are 0, 1, or 4; only narrow values"]
