We are given that \( p \) is a positive multiple of 6 and \( p^3 < 8000 \). To find the largest possible value of \( p \), we first determine the cube root of 8000:

["Finding the Largest Positive Multiple of 6 Less Than the Cube Root of 8000", "When solving mathematical constraints like ( p ) being a positive multiple of 6 and satisfying ( p^3 < 8000 ), finding the largest possible value of ( p ) requires careful calculation. To start, we determine the cube root of 8000, since it provides a clear upper boundary for ( p ).", "We first compute:\n[\n\sqrt[3]{8000} = 20\n]\nbecause ( 20 \ imes 20 \ imes 20 = 8000 ). This means ( p ) must be strictly less than 20. Since ( p ) is a positive multiple of 6, we list the positive multiples of 6 that are less than 20:\n[\n6,\ 12,\ 18\n]\nAmong these, the largest value is ( 18 ). Checking:\n[\n18^3 = 5832,\quad \ ext{which is less than } 8000\n]\nBut ( 24^3 = 13824 > 8000 ), so ( 18 ) is indeed the largest valid multiple.", "Thus, the largest possible value of ( p ) satisfying ( p ) is a positive multiple of 6 and ( p^3 < 8000 ) is:", "[\n\boxed{18}\n]", "This solution highlights how understanding cube roots helps narrow down candidate values efficiently, especially when working with multiplicative constraints like multiples of 6. By combining number theory with inequality reasoning, we quickly identify the optimum solution.", "---", "Keywords: largest multiple of 6 less than 20, cube root of 8000, ( p^3 < 8000 ), positive integer multiple, math problem solution, constraints and inequalities", "Meta Description:\nFind the largest positive multiple of 6 such that ( p^3 < 8000 ). Learn step-by-step how to determine ( p = 18 ) through cube root analysis and valid multiple checks."]