IMO Shortlist 2007 problem N7
For a prime
and a given integer
let
denote the exponent of
in the prime factorisation of
. Given
and
a set of
primes, show that there are infinitely many positive integers
such that
for all
.
Author: Tejaswi Navilarekkallu, India
%V0
For a prime $p$ and a given integer $n$ let $\nu_p(n)$ denote the exponent of $p$ in the prime factorisation of $n!$. Given $d \in \mathbb{N}$ and $\{p_1,p_2,\ldots,p_k\}$ a set of $k$ primes, show that there are infinitely many positive integers $n$ such that $d|\nu_{p_i}(n)$ for all $1 \leq i \leq k$.
Author: Tejaswi Navilarekkallu, India
Source: Međunarodna matematička olimpijada, shortlist 2007
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