IMO Shortlist 2007 problem N7
Dodao/la:
arhiva2. travnja 2012. For a prime
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
and a given integer
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
let
![\nu_p(n)](/media/m/9/a/0/9a05acbb0b47ab2a3f3d71f2d79f22d7.png)
denote the exponent of
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
in the prime factorisation of
![n!](/media/m/5/e/9/5e9bb819f1bfbf465700f6bc8831a1c7.png)
. Given
![d \in \mathbb{N}](/media/m/5/6/f/56f7bf57a2515f15b14e5daf24c2be8f.png)
and
![\{p_1,p_2,\ldots,p_k\}](/media/m/4/8/5/485329c9119d2ae1d765c11dddf8ce2d.png)
a set of
![k](/media/m/f/1/3/f135be660b73381aa6bec048f0f79afc.png)
primes, show that there are infinitely many positive integers
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
such that
![d|\nu_{p_i}(n)](/media/m/c/c/7/cc7d5359d311ccdfcbf9d6bae6d661db.png)
for all
![1 \leq i \leq k](/media/m/8/0/0/80006023e32f0330afb09eb307118f82.png)
.
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
Izvor: Međunarodna matematička olimpijada, shortlist 2007