Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

How to show that a real number has a finite decimal representation (one that ends with an infinite sequence of zeros) if and only if it can be represented as a rational number m/n where n has no prime divisors other than 2 and 5.

share|improve this question

1 Answer 1

up vote 0 down vote accepted

It is the case iff it is written $$\frac a {10^n}$$ for two integers $(a,n)$. so this is a rational number with proper form$$\frac pq$$ with $\gcd(p,q) = 1$.

As $10^n p = aq$ and $\gcd(p,q) = 1$, use Gauss theorem yields $q|10^n$.

share|improve this answer
    
Can you explain Gauss theorem because I've never seen this. This there any other way to solve this without that theorem? –  ayv2 Mar 3 at 22:16
    
Gauss theorem: $a|bc$ and $\gcd(a,c) = 1\Rightarrow a|b$. This is how you get unicity of the prime number factorization. –  mookid Mar 3 at 22:52

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.