BdMO National Higher Secondary 2019/8

Discussion on Bangladesh Mathematical Olympiad (BdMO) National
User avatar
samiul_samin
Posts: 1004
Joined: Sat Dec 09, 2017 1:32 pm

BdMO National Higher Secondary 2019/8

Unread post by samiul_samin » Mon Mar 04, 2019 9:44 am

The set of natural numbers $\mathbb{N}$ are partitioned into a finite number of subsets.Prove that there exists a subset of $S$ so that for any natural numbers $n$,there are infinitely many multiples of $n$ in $S$.

User avatar
samiul_samin
Posts: 1004
Joined: Sat Dec 09, 2017 1:32 pm

Re: BdMO National Higher Secondary 2019/8

Unread post by samiul_samin » Sat Mar 09, 2019 10:12 am

Hint
Use Infinite pigeon hole principle
Short Solution
Infinite Pigeon =infinite multiple
Finite hole=finite subset.
So,by using infinte pigeon hole principle we can say that there are infinitely many multiples of $n$ in $S$.

It can also be proved by contradiction as finite×finite not equals to infinite.

soyeb pervez jim
Posts: 21
Joined: Sat Jan 28, 2017 11:06 pm

Re: BdMO National Higher Secondary 2019/8

Unread post by soyeb pervez jim » Sun Mar 17, 2019 7:49 pm

May be this answer is not correct as the question asked to prove that there exists a subset $S$ such that in $S$ there are infinitely many multiples of any natural number $n$.

here you have proven for a natural number $n$ there is a subset which have infinite multiple of $n$. But you have to prove in subset $S$ there are infinity many multiples of any natural number $n$

Post Reply