Question: In a drug trial, a compound is tested at doses corresponding to the cube of a number. What is the smallest positive integer whose cube ends in $ 888 $?
["In a drug trial, a compound is tested at doses corresponding to the cube of a number. What is the smallest positive integer whose cube ends in 888?", "For users exploring breakthroughs in medicine, a quiet but growing curiosity is shaping digital searches: What integer, when cubed, ends in the digits 888? This question reflects broader conversations about precision in drug development—how mathematical patterns intersect with scientific testing, especially in clinical trials. As data-driven health trends accelerate, identifying exact numerical solutions mirrors real-world algorithms used in pharmacological modeling.", "Why This Question Matters in the US Quest for Medical Precision \nRising interest in drug efficacy, safety thresholds, and computational modeling fuels attention on unique number properties like cube endings. Shoppers, caregivers, and researchers alike seek clarity in complex scientific processes—this query epitomizes that demand. The search for the smallest cube ending in 888 exemplifies how everyday curiosity aligns with advanced pharmaceutical analysis. It’s not just a numeral puzzle, but a gateway into how math aids real-world medicine.", "How to Crack the Cube That Ends in 888 \nFinding the smallest positive integer \( n \) such that \( n^3 \) ends in 888 hinges on examining cube endings modulo 1000. Rather than brute force, a smarter approach checks the cube’s final three digits systematically. Since \( n^3 \mod 1000 = 888 \), we test integers incrementally—leveraging modular arithmetic insight not to exploit performance, but to uncover patterns hidden in number theory. The process reveals one honest answer: 942.", "Here’s how it works: \n- Compute \( 942^3 \) modulo 1000 step-by-step: \n First, calculate \( 942^2 = 887,364 \), then \( 942 \cdot 887,364 \mod 1000 \). \n Last three digits of \( 942 \ imes 364 = 342,648 \mod 1000 = 648. \) \n Then \( 942 \ imes 648 = 610,536 \mod 1000 = 536. \) — close but not 888. \n- Through deeper testing, refinement identifies \( 192^3 = 7,077,888 \), revealing \( 888 \) at the end.", "Verification confirms \( 192^3 = 7,077,888 \), the smallest such number.", "Common Questions About Cube Ends in 888", "H3: How Is It Possible for a Cube to End in 888? \nMathematically, only certain residues modulo 1000 yield cube endings like 888. Testing all integers shows no smaller positive n satisfies \( n^3 \equiv 888 \pmod{"]
