Art of Problem Solving Forum

Do you have what it takes to be the next brilliant trader, researcher, or developer at Jane Street Capital? Find out in the Careers in Mathematics Forum.

The time now is Sat Dec 05, 2009 7:48 am
All times are UTC - 8

View posts since last visit
View unanswered posts

Home » Olympiad Section » Number Theory » Number Theory Open Questions

d is a power of 10
Moderators: High School Olympiad Moderators, amfulger, Arne, darij grinberg, freemind, harazi, Megus, N.T.TUAN, orl, pbornsztein, ZetaX

2 Posts • Page 1 of 1

Author

Message

b555
Poincare Conjecture

Offline
Joined: 07 Mar 2009
Posts: 221

To rate posts you must be logged in

d is a power of 10
tornament of town 2001 --q5

Let a and d be positie integers. for any positive integer `n`, the number a+nd conntains a block of consecutive digits which constitute the number `n`. Prove that `d` is a power of 10.

Posted: Wed Oct 07, 2009 10:20 am

Profile PM

pco
Navier-Stokes Equations

Online
Joined: 28 Apr 2007
Posts: 2015
Location: Paris (France)

To rate posts you must be logged in

Re: d is a power of 10
tornament of town 2001 --q5

b555 wrote:

Let a and d be positie integers. for any positive integer `n`, the number a+nd conntains a block of consecutive digits which constitute the number `n`. Prove that `d` is a power of 10.

Let $d=\overline{d_1xx...xxd_20...0}$ with $d_1\neq 0,d_2\neq 0$ , $q$ trailing zeroes and $p+q$ total digits.
Let $a=\overline{d_3xx...xx}$ with $r$ total digits.

Let $k=1000+\max(p+q,r)$

Let $N=a+10^kd$ $=\overline{d_1xx...xxd_200...00d_3xx...xx}$
The left part ( $\overline{d_1xx...xxd_2}$ ) has $p<k$ digits and so cant contain the representation of $10^k$
The right part ( $\overline{d_3xx...xx}$ ) has $r<k$ digits and so cant contain the representation of $10^k$
The middle part contains $k+q-r$ zeroes

In order the middle part contains the representation of $10^k$ , we need $d_2=1$ and $k+q-r\geq k$ $\iff$ $q\geq r$

Let then $M=a+(10^k+1)d=N+d$ $=\overline{d_1xx...xxd_200...00xxx...xx}$
The left part ( $\overline{d_1xx...xxd_2}$ ) has $p<k$ digits and so cant contain the representation of $10^k+1$
The right part is $a+d$ and so has $p+q<k$ or $p+q+1<k$ digits and so cant contain the representation of $10^k+1$
The middle part contains $k+q-(p+q)=k-p$ or $k+q-(p+q+1)=k-p-1$ zeroes.

In order the middle part contains the representation of $10^k+1$ , we need $k-p\geq k-1$ or $k-p-1\geq k-1$ and so $p\leq 1$

So $p=1$ and $d_1=d_2=1$ and so $d=\overline{10...0}=10^q$

Q.E.D.

_________________
Patrick

Posted: Thu Oct 08, 2009 11:01 am

Profile PM

Display posts from previous: Sort by:

2 Posts • Page 1 of 1

Home » Olympiad Section » Number Theory » Number Theory Open Questions

Jump to:

You cannot post new topics in this forum
You cannot reply to topics in this forum
You cannot edit your posts in this forum
You cannot delete your posts in this forum
You cannot vote in polls in this forum
You cannot attach files in this forum
You can download files in this forum
You cannot post calendar events in this forum