Challenge VII: A bit of number theory solved by Boorglar

[micromass]

This new challenge was suggested by jostpuur. It is rather number theoretic.

Assume that $q\in \mathbb{Q}$ is an arbitrary positive rational number. Does there exist a natural number $L\in \mathbb{N}$ such that

$$Lq=99…9900…00$$

with some amounts of nines and zeros? Prove or find a counterexample.

 

[Boorglar]

Spoiler

So we want to get a number of the form $L \frac{a}{b} = 10^m(10^n-1)$.
Write $a = 2^x5^yd$, where d is relatively prime with 10.

Then $L = \frac{10^m}{2^x5^y} \frac{10^n-1}{d}$.
The left fraction is obviously an integer if we choose m larger than max( x, y ).
The right fraction can be made an integer since d is relatively prime with 10, and therefore 10 is in the multiplicative group modulo d. Let n be the order of 10 in U(d), then $10^n-1$ is divisible by d so L is an integer.

 

[micromass]

Well, that was fast.

 

[Boorglar]

Yeah I tend to be good at those types of problems haha.
Ah by the way I forgot the b multiplying the fractions but it doesn't really matter.