IMO 1971 LL YUG53

Denote by xn(p) the multiplicity of the prime p in the canonical

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IMO 1971 LL YUG53

Origin: YUG

Problem

Denote by xn(p) the multiplicity of the prime p in the canonical representation of the number n! as a product of primes. Prove that xn(p) n < p−1 and limn\to\infty xn(p) n

p−1.