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I recently stumbled upon this really interesting problem:

Suppose we have a fraction $\frac{a}{b}$ where $a,b \in \mathbb{N}$ and we know that the decimal fraction of $\frac{a}{b}$ has the numerical sequence $7143$ somewhere in the decimal place. Show that $b > 1250 $.

This question is part of der Bundeswettbewerb Mathematik 2015, zweite Rund. The competition ended September 1st, 2015.

Any kind of help will be appreciated!

Peter Woolfitt
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3 Answers3

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The trick is to realize $7\times0.7143=5.0001$.

First let's multiply by $10^n$ to shift the decimal to the right - hence for some integers $k$ and $n$ and real number $c\in[0,1)$ we can write

$$\frac{10^na}{b}=k+0.7143+0.0001c$$

Now multiply by our magic number!

$$7\times\frac{10^na}{b}=7k+5.0001+0.0007c$$

Hence

$$7(10^na-kb)-5b=(0.0001+0.0007c)b$$

The left hand side is an integer, so there exists $m\in\mathbb{Z}$ such that

$$m=(0.0001+0.0007c)b$$

Since $0\le c<1$, we have

$$0.0001 b\le m<0.0008b$$

Additionally, since $b\in\mathbb{N}$, we have $$0<0.0001b\le m$$ so because $m$ is an integer, $$1\le m<0.0008b$$ giving $1250<b$ as desired.

Peter Woolfitt
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    That's definitely the most elementary approach. :) – Thomas Andrews May 29 '15 at 12:15
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    WOW! What a beautiful solution. –  May 29 '15 at 12:43
  • @PeterWoolfitt I was wondering if we can solve this by reverse proofing. That is, we start with assuming b is definitely greater than 1250 and arrive to a fraction with 7143 in decimal. Can that be done? I tried doing that way but was lost. – MonK May 29 '15 at 12:56
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    @MonK I'm not sure exactly what you mean. From that it sounds like you trying to prove the converse - which isn't in general true (not every number greater than $1250$ yields a fraction with $7143$ in the decimal, like $\frac{1}{10000}$). – Peter Woolfitt May 29 '15 at 13:02
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    Proof from the book :) – user135988 May 29 '15 at 13:04
  • @Silenttiffy, ThomasAndrews: Thanks! I can't claim all the credit for this idea though - I had a conversation with user JimmyK4542 a couple years ago where he demonstrated that some numbers can be easily factored (by hand) if they are first $\textit{multiplied}$ (try factoring $142857$ vs. factoring $142857\times 7$ (you might recognize $0.142857$ as close to the decimal expansion of $\frac{1}{7}$)). – Peter Woolfitt May 29 '15 at 13:07
  • Very nice proof but I fail to grasp how it could be generalized to a generic sequence of numbers. Also, can it only be used to prove that a given threshold is true, or can it also be used to find the minimum value for b? – UncleZeiv May 29 '15 at 14:08
  • The method generalizes for any sequence of digits which has a (smallish) multiple which is near a (smallish multiple of a) power of $10$. So for example the same approach would work for $857143$ since $7\times857143=6000001$. The method can only be used to prove a threshold, but that is to be expected because we are using rather loose estimates (in the sense that we don't care what the following decimal places could be). – Peter Woolfitt Aug 08 '16 at 07:10
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First multiply a by a power of 10 such that 7143 comes right after the decimal point, and then subtract a multiple of b such that the decimal fraction starts with $0.7143$. The claim we need to prove is then that the interval $[0.7143,0.7144)$ does not contain any rational with a denominator $\le 1250$.

The continued fractions for $0.7143$ and $0.7144$ are $[0;1,2,1,1,1428]$ and $[0;1,2,1,1,178]$, so the continued fraction expansion for every number in this interval will have the form $[0;1,2,1,1,n,\ldots]$ where $178\le n\le 1428$ and there may or may not be more terms after the $n$.

If we work out $[0;1,2,1,1,n]$ as an ordinary fraction we get $\frac{5n+3}{7n+4}$, which (being a continued fraction approximant) is always in lowest terms.

When $n>178$ the denominator of this is at least $7\cdot 179+4=1257$. Since the denominators of continued fraction approximants always increase, we get $b \ge 1257$.

On the other hand we might have $n=178$ if there is a term after $n$ in the continued fraction expansion. But the simplest fraction we can then make corresponds to $n=178.5$, which would make the next approximant $\frac{5\cdot 178.5+3}{7\cdot 178.5+4}=\frac{1791}{2507}$, also with a large denominator.

hmakholm left over Monica
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By multiplying $a$ with a power of $10$ if necessary, we may assume that $7143$ occurs immediately after the decimal point, and by subtracting a multiple of $b$ from $a$ we may assume that in fact $\frac ab=0.7143\ldots$. Then $$ \frac57=0.7142857\ldots <\frac ab <0.7144 = \frac{893}{1250}$$ Hence the numerators of the difference fractions $\frac ab-\frac 57=\frac{7a-5b}{7b}$ and $\frac{893}{1250}-\frac ab=\frac{893b-1250a}{1250b}$ are positive integers, i.e., $$ \begin{align}7a-\hphantom{88}5b&\ge 1\\-1250a+893b&\ge1\end{align}$$ Then $1250$ times the first inequality plus $7$ times the second inequality eliminates $a$, which gives us $$ (-5\cdot 1250+7\cdot 893) b\ge 1250+7,$$ i.e., $$ b\ge 1257.$$

By the way, $1257$ is the best bound since $$\frac{5+893}{7+1250}=\frac{898}{1257}=0.714399363564\ldots$$

Apass.Jack
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Hagen von Eitzen
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