Number theory · Gcd lcm · Inequalities
Let \(a\) and \(b\) be arbitrary natural numbers, let \(M\) be their least common multiple and \(D\) their greatest common divisor. Prove that \[ a^n + b^n \le M^n + D^n \] holds for every natural number \(n\).
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Serbian Municipal Competition 1997, high school grade I, problem 2. Organized by the Mathematical Society of Serbia (DMS).